Analysis of the scalar, axialvector, vector, tensor doubly charmed tetraquark states with QCD sum rules

Zhi-Gang Wang 111E-mail: zgwang@aliyun.com. , Ze-Hui Yan

Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

Abstract

In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, vector, tensor doubly charmed tetraquark states, and study them with QCD sum rules systematically by carrying out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way, the predicted masses can be confronted to the experimental data in the future. We can search for those doubly charmed tetraquark states in the Okubo-Zweig-Iizuka super-allowed strong decays to the charmed meson pairs.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

1 Introduction

Recently, the LHCb collaboration observed the doubly charmed baryon Ξcc++superscriptsubscriptΞ𝑐𝑐absent\Xi_{cc}^{++} in the Λc+Kπ+π+superscriptsubscriptΛ𝑐superscript𝐾superscript𝜋superscript𝜋\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+} mass spectrum, and obtained the mass MΞcc++=3621.40±0.72±0.27±0.14MeVsubscript𝑀superscriptsubscriptΞ𝑐𝑐absentplus-or-minus3621.400.720.270.14MeVM_{\Xi_{cc}^{++}}=3621.40\pm 0.72\pm 0.27\pm 0.14\,\rm{MeV}, but did not measure the spin [1]. The doubly heavy baryon configuration QQq𝑄𝑄𝑞QQq is very similar to the heavy-light meson Q¯q¯𝑄𝑞\bar{Q}q, where we have a doubly heavy diquark QQ𝑄𝑄QQ instead of a heavy antiquark Q¯¯𝑄\bar{Q} in color antitriplet. The attractive interaction induced by one-gluon exchange favors formation of the diquarks in color antitriplet [2], the favored configurations are the scalar (Cγ5𝐶subscript𝛾5C\gamma_{5}) and axialvector (Cγμ𝐶subscript𝛾𝜇C\gamma_{\mu}) diquark states [3, 4]. For the cc𝑐𝑐cc quark system, only the axialvector diquark εijkcjTCγμcksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑗𝐶subscript𝛾𝜇subscript𝑐𝑘\varepsilon^{ijk}c^{T}_{j}C\gamma_{\mu}c_{k} and tensor diquark εijkcjTCσμνcksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑗𝐶subscript𝜎𝜇𝜈subscript𝑐𝑘\varepsilon^{ijk}c^{T}_{j}C\sigma_{\mu\nu}c_{k} survive due to the Fermi-Dirac statistics, the axialvector diquark εijkcjTCγμcksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑗𝐶subscript𝛾𝜇subscript𝑐𝑘\varepsilon^{ijk}c^{T}_{j}C\gamma_{\mu}c_{k} is more stable than the tensor diquark εijkcjTCσμνcksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑗𝐶subscript𝜎𝜇𝜈subscript𝑐𝑘\varepsilon^{ijk}c^{T}_{j}C\sigma_{\mu\nu}c_{k}, the observation of the Ξcc++superscriptsubscriptΞ𝑐𝑐absent\Xi_{cc}^{++} indicates that there exists strong correlation between the two charm quarks. We can take the diquark εijkciTCγμcjsuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑖𝐶subscript𝛾𝜇subscript𝑐𝑗\varepsilon^{ijk}c^{T}_{i}C\gamma_{\mu}c_{j} as basic constituent to construct the spin 1212\frac{1}{2} current

JΞcc(x)subscript𝐽subscriptΞ𝑐𝑐𝑥\displaystyle J_{\Xi_{cc}}(x) =\displaystyle= εijkciT(x)Cγμcj(x)γ5γμuk(x),superscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑖𝑥𝐶subscript𝛾𝜇subscript𝑐𝑗𝑥subscript𝛾5superscript𝛾𝜇subscript𝑢𝑘𝑥\displaystyle\varepsilon^{ijk}c^{T}_{i}(x)C\gamma_{\mu}c_{j}(x)\gamma_{5}\gamma^{\mu}u_{k}(x)\,, (1)

or the spin 3232\frac{3}{2} current

JΞccμ(x)superscriptsubscript𝐽subscriptΞ𝑐𝑐𝜇𝑥\displaystyle J_{\Xi_{cc}}^{\mu}(x) =\displaystyle= εijkciT(x)Cγμcj(x)uk(x),superscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑖𝑥𝐶superscript𝛾𝜇subscript𝑐𝑗𝑥subscript𝑢𝑘𝑥\displaystyle\varepsilon^{ijk}c^{T}_{i}(x)C\gamma^{\mu}c_{j}(x)u_{k}(x)\,, (2)

to study the Ξcc++superscriptsubscriptΞ𝑐𝑐absent\Xi_{cc}^{++} with the QCD sum rules [5].

The doubly heavy tetraquark state QQq¯q¯𝑄𝑄¯𝑞superscript¯𝑞QQ\bar{q}\bar{q}^{\prime} is very similar to the doubly heavy baryon state QQq𝑄𝑄𝑞QQq, where we have a light antidiquark q¯q¯¯𝑞superscript¯𝑞\bar{q}\bar{q}^{\prime} instead of a light quark q𝑞q in color triplet. The observation of the Ξcc++superscriptsubscriptΞ𝑐𝑐absent\Xi_{cc}^{++} provides the crucial experimental input on the strong correlation between the two charm quarks, which may shed light on the spectroscopy of the doubly charmed tetraquark states. An axialvector doubly charmed diquark state can combine with an axialvector or scalar light antidiquark state to form a compact doubly charmed tetraquark state, it is interesting to revisit this subject with the QCD sum rules. The QCD sum rules is a powerful theoretical tool in studying the ground state hadrons, and has given many successful descriptions of the hadronic parameters on the phenomenological side [6, 7]. Up to now, no experimental candidates for the doubly charmed tetraquark states ccq¯q¯𝑐𝑐¯𝑞superscript¯𝑞cc\bar{q}\bar{q}^{\prime} or qqc¯c¯𝑞superscript𝑞¯𝑐¯𝑐qq^{\prime}\bar{c}\bar{c} have been observed. There have been several works on the doubly heavy tetraquark states, such as potential quark models [8, 9], QCD sum rules [10, 11, 12], heavy quark symmetry [13, 14], lattice QCD [15, 16], etc.

In previous work, we study the axialvector doubly heavy tetraquark states, which consist of an axialvector diquark εijkQjTCγμQksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑄𝑇𝑗𝐶subscript𝛾𝜇subscript𝑄𝑘\varepsilon^{ijk}Q^{T}_{j}C\gamma_{\mu}Q_{k} and a scalar antidiquark εijkq¯jTγ5Cq¯ksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript¯𝑞𝑇𝑗subscript𝛾5𝐶subscriptsuperscript¯𝑞𝑘\varepsilon^{ijk}\bar{q}^{T}_{j}\gamma_{5}C\bar{q}^{\prime}_{k}, with the QCD sum rules in details by taking into account the energy scale dependence of the QCD spectral densities [17]. In this article, we choose the axialvector diquark εijkcjTCγμcksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript𝑐𝑇𝑗𝐶subscript𝛾𝜇subscript𝑐𝑘\varepsilon^{ijk}c^{T}_{j}C\gamma_{\mu}c_{k} and axialvector antidiquark εijkq¯jTγμCq¯ksuperscript𝜀𝑖𝑗𝑘subscriptsuperscript¯𝑞𝑇𝑗subscript𝛾𝜇𝐶subscriptsuperscript¯𝑞𝑘\varepsilon^{ijk}\bar{q}^{T}_{j}\gamma_{\mu}C\bar{q}^{\prime}_{k} to construct the currents to interpolate the doubly charmed tetraquark states with the spin-parity JP=0+, 1±, 2+superscript𝐽𝑃superscript0superscript1plus-or-minussuperscript2J^{P}=0^{+},\,1^{\pm},\,2^{+}, and study them with the QCD sum rules systematically by taking into account the contributions of the vacuum condensates up to dimension 10 in a consistent way in the operator product expansion.

The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the doubly charmed tetraquark states in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.

2 The QCD sum rules for the doubly charmed tetraquark states

In the following, we write down the two-point correlation functions Π0(p)subscriptΠ0𝑝\Pi_{0}(p), Πμναβ;1(p)subscriptΠ𝜇𝜈𝛼𝛽1𝑝\Pi_{\mu\nu\alpha\beta;1}(p) and Πμναβ;2(p)subscriptΠ𝜇𝜈𝛼𝛽2𝑝\Pi_{\mu\nu\alpha\beta;2}(p) in the QCD sum rules,

Π0(p)subscriptΠ0𝑝\displaystyle\Pi_{0}(p) =\displaystyle= id4xeipx0|T{J0(x)J0(0)}|0,𝑖superscript𝑑4𝑥superscript𝑒𝑖𝑝𝑥quantum-operator-product0𝑇subscript𝐽0𝑥superscriptsubscript𝐽000\displaystyle i\int d^{4}xe^{ip\cdot x}\langle 0|T\left\{J_{0}(x)J_{0}^{\dagger}(0)\right\}|0\rangle\,,
Πμναβ;1(p)subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle\Pi_{\mu\nu\alpha\beta;1}(p) =\displaystyle= id4xeipx0|T{Jμν;1(x)Jαβ;1(0)}|0,𝑖superscript𝑑4𝑥superscript𝑒𝑖𝑝𝑥quantum-operator-product0𝑇subscript𝐽𝜇𝜈1𝑥superscriptsubscript𝐽𝛼𝛽100\displaystyle i\int d^{4}xe^{ip\cdot x}\langle 0|T\left\{J_{\mu\nu;1}(x)J_{\alpha\beta;1}^{\dagger}(0)\right\}|0\rangle\,,
Πμναβ;2(p)subscriptΠ𝜇𝜈𝛼𝛽2𝑝\displaystyle\Pi_{\mu\nu\alpha\beta;2}(p) =\displaystyle= id4xeipx0|T{Jμν;2(x)Jαβ;2(0)}|0,𝑖superscript𝑑4𝑥superscript𝑒𝑖𝑝𝑥quantum-operator-product0𝑇subscript𝐽𝜇𝜈2𝑥superscriptsubscript𝐽𝛼𝛽200\displaystyle i\int d^{4}xe^{ip\cdot x}\langle 0|T\left\{J_{\mu\nu;2}(x)J_{\alpha\beta;2}^{\dagger}(0)\right\}|0\rangle\,, (3)

where J0(x)=Ju¯d¯;0(x),Ju¯s¯;0(x),Js¯s¯;0(x)subscript𝐽0𝑥subscript𝐽¯𝑢¯𝑑0𝑥subscript𝐽¯𝑢¯𝑠0𝑥subscript𝐽¯𝑠¯𝑠0𝑥J_{0}(x)=J_{\bar{u}\bar{d};0}(x),\,J_{\bar{u}\bar{s};0}(x),\,J_{\bar{s}\bar{s};0}(x), Jμν;1(x)=Jμν;u¯d¯;1(x),Jμν;u¯s¯;1(x),Jμν;s¯s¯;1(x)subscript𝐽𝜇𝜈1𝑥subscript𝐽𝜇𝜈¯𝑢¯𝑑1𝑥subscript𝐽𝜇𝜈¯𝑢¯𝑠1𝑥subscript𝐽𝜇𝜈¯𝑠¯𝑠1𝑥J_{\mu\nu;1}(x)=J_{\mu\nu;\bar{u}\bar{d};1}(x),\,J_{\mu\nu;\bar{u}\bar{s};1}(x),\,J_{\mu\nu;\bar{s}\bar{s};1}(x), Jμν;2(x)=Jμν;u¯d¯;2(x),Jμν;u¯s¯;2(x),Jμν;s¯s¯;2(x)subscript𝐽𝜇𝜈2𝑥subscript𝐽𝜇𝜈¯𝑢¯𝑑2𝑥subscript𝐽𝜇𝜈¯𝑢¯𝑠2𝑥subscript𝐽𝜇𝜈¯𝑠¯𝑠2𝑥J_{\mu\nu;2}(x)=J_{\mu\nu;\bar{u}\bar{d};2}(x),\,J_{\mu\nu;\bar{u}\bar{s};2}(x),\,J_{\mu\nu;\bar{s}\bar{s};2}(x),

Ju¯d¯;0(x)subscript𝐽¯𝑢¯𝑑0𝑥\displaystyle J_{\bar{u}\bar{d};0}(x) =\displaystyle= εijkεimncjT(x)Cγμck(x)u¯m(x)γμCd¯nT(x),superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥superscript𝛾𝜇𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma^{\mu}C\bar{d}^{T}_{n}(x)\,,
Ju¯s¯;0(x)subscript𝐽¯𝑢¯𝑠0𝑥\displaystyle J_{\bar{u}\bar{s};0}(x) =\displaystyle= εijkεimncjT(x)Cγμck(x)u¯m(x)γμCs¯nT(x),superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥superscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma^{\mu}C\bar{s}^{T}_{n}(x)\,,
Js¯s¯;0(x)subscript𝐽¯𝑠¯𝑠0𝑥\displaystyle J_{\bar{s}\bar{s};0}(x) =\displaystyle= εijkεimncjT(x)Cγμck(x)s¯m(x)γμCs¯nT(x),superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑠𝑚𝑥superscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{s}_{m}(x)\gamma^{\mu}C\bar{s}^{T}_{n}(x)\,,
Jμν;u¯d¯;1(x)subscript𝐽𝜇𝜈¯𝑢¯𝑑1𝑥\displaystyle J_{\mu\nu;\bar{u}\bar{d};1}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)u¯m(x)γνCd¯nT(x)cjT(x)Cγνck(x)u¯m(x)γμCd¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\nu}C\bar{d}^{T}_{n}(x)-c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\mu}C\bar{d}^{T}_{n}(x)\right]\,,
Jμν;u¯s¯;1(x)subscript𝐽𝜇𝜈¯𝑢¯𝑠1𝑥\displaystyle J_{\mu\nu;\bar{u}\bar{s};1}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)u¯m(x)γνCs¯nT(x)cjT(x)Cγνck(x)u¯m(x)γμCs¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\nu}C\bar{s}^{T}_{n}(x)-c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\mu}C\bar{s}^{T}_{n}(x)\right]\,,
Jμν;s¯s¯;1(x)subscript𝐽𝜇𝜈¯𝑠¯𝑠1𝑥\displaystyle J_{\mu\nu;\bar{s}\bar{s};1}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)s¯m(x)γνCs¯nT(x)cjT(x)Cγνck(x)s¯m(x)γμCs¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑠𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑠𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{s}_{m}(x)\gamma_{\nu}C\bar{s}^{T}_{n}(x)-c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{s}_{m}(x)\gamma_{\mu}C\bar{s}^{T}_{n}(x)\right]\,,
Jμν;u¯d¯;2(x)subscript𝐽𝜇𝜈¯𝑢¯𝑑2𝑥\displaystyle J_{\mu\nu;\bar{u}\bar{d};2}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)u¯m(x)γνCd¯nT(x)+cjT(x)Cγνck(x)u¯m(x)γμCd¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\nu}C\bar{d}^{T}_{n}(x)+c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\mu}C\bar{d}^{T}_{n}(x)\right]\,,
Jμν;u¯s¯;2(x)subscript𝐽𝜇𝜈¯𝑢¯𝑠2𝑥\displaystyle J_{\mu\nu;\bar{u}\bar{s};2}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)u¯m(x)γνCs¯nT(x)+cjT(x)Cγνck(x)u¯m(x)γμCs¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\nu}C\bar{s}^{T}_{n}(x)+c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\mu}C\bar{s}^{T}_{n}(x)\right]\,,
Jμν;s¯s¯;2(x)subscript𝐽𝜇𝜈¯𝑠¯𝑠2𝑥\displaystyle J_{\mu\nu;\bar{s}\bar{s};2}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)s¯m(x)γνCs¯nT(x)+cjT(x)Cγνck(x)s¯m(x)γμCs¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑠𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑠𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑠𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{s}_{m}(x)\gamma_{\nu}C\bar{s}^{T}_{n}(x)+c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{s}_{m}(x)\gamma_{\mu}C\bar{s}^{T}_{n}(x)\right]\,,

the i𝑖i, j𝑗j, k𝑘k, m𝑚m, n𝑛n are color indexes, the C𝐶C is the charge conjugation matrix. We choose the currents J0(x)subscript𝐽0𝑥J_{0}(x), Jμν;1(x)subscript𝐽𝜇𝜈1𝑥J_{\mu\nu;1}(x) and Jμν;2(x)subscript𝐽𝜇𝜈2𝑥J_{\mu\nu;2}(x) to interpolate the spin-parity JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}, 1±superscript1plus-or-minus1^{\pm} and 2+superscript22^{+} doubly charmed tetraquark states, respectively.

On the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J0(x)subscript𝐽0𝑥J_{0}(x), Jμν;1(x)subscript𝐽𝜇𝜈1𝑥J_{\mu\nu;1}(x) and Jμν;2(x)subscript𝐽𝜇𝜈2𝑥J_{\mu\nu;2}(x) into the correlation functions Π0(p)subscriptΠ0𝑝\Pi_{0}(p), Πμναβ;1(p)subscriptΠ𝜇𝜈𝛼𝛽1𝑝\Pi_{\mu\nu\alpha\beta;1}(p) and Πμναβ;2(p)subscriptΠ𝜇𝜈𝛼𝛽2𝑝\Pi_{\mu\nu\alpha\beta;2}(p) respectively to obtain the hadronic representation [6, 7], and isolate the ground state contributions,

Π0(p)subscriptΠ0𝑝\displaystyle\Pi_{0}(p) =\displaystyle= λZ2MZ2p2+,superscriptsubscript𝜆𝑍2superscriptsubscript𝑀𝑍2superscript𝑝2\displaystyle\frac{\lambda_{Z}^{2}}{M_{Z}^{2}-p^{2}}+\cdots\,\,, (5)
=\displaystyle= Π0(p2),subscriptΠ0superscript𝑝2\displaystyle\Pi_{0}(p^{2})\,\,,
Πμναβ;1(p)subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle\Pi_{\mu\nu\alpha\beta;1}(p) =\displaystyle= λZ2MZ2p2(p2gμαgνβp2gμβgναgμαpνpβgνβpμpα+gμβpνpα+gναpμpβ)superscriptsubscript𝜆𝑍2superscriptsubscript𝑀𝑍2superscript𝑝2superscript𝑝2subscript𝑔𝜇𝛼subscript𝑔𝜈𝛽superscript𝑝2subscript𝑔𝜇𝛽subscript𝑔𝜈𝛼subscript𝑔𝜇𝛼subscript𝑝𝜈subscript𝑝𝛽subscript𝑔𝜈𝛽subscript𝑝𝜇subscript𝑝𝛼subscript𝑔𝜇𝛽subscript𝑝𝜈subscript𝑝𝛼subscript𝑔𝜈𝛼subscript𝑝𝜇subscript𝑝𝛽\displaystyle\frac{\lambda_{Z}^{2}}{M_{Z}^{2}-p^{2}}\left(p^{2}g_{\mu\alpha}g_{\nu\beta}-p^{2}g_{\mu\beta}g_{\nu\alpha}-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)
+λY2MY2p2(gμαpνpβgνβpμpα+gμβpνpα+gναpμpβ)+,superscriptsubscript𝜆𝑌2superscriptsubscript𝑀𝑌2superscript𝑝2subscript𝑔𝜇𝛼subscript𝑝𝜈subscript𝑝𝛽subscript𝑔𝜈𝛽subscript𝑝𝜇subscript𝑝𝛼subscript𝑔𝜇𝛽subscript𝑝𝜈subscript𝑝𝛼subscript𝑔𝜈𝛼subscript𝑝𝜇subscript𝑝𝛽\displaystyle+\frac{\lambda_{Y}^{2}}{M_{Y}^{2}-p^{2}}\left(-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)+\cdots\,\,,
=\displaystyle= ΠZ(p2)(p2gμαgνβp2gμβgναgμαpνpβgνβpμpα+gμβpνpα+gναpμpβ)subscriptΠ𝑍superscript𝑝2superscript𝑝2subscript𝑔𝜇𝛼subscript𝑔𝜈𝛽superscript𝑝2subscript𝑔𝜇𝛽subscript𝑔𝜈𝛼subscript𝑔𝜇𝛼subscript𝑝𝜈subscript𝑝𝛽subscript𝑔𝜈𝛽subscript𝑝𝜇subscript𝑝𝛼subscript𝑔𝜇𝛽subscript𝑝𝜈subscript𝑝𝛼subscript𝑔𝜈𝛼subscript𝑝𝜇subscript𝑝𝛽\displaystyle\Pi_{Z}(p^{2})\left(p^{2}g_{\mu\alpha}g_{\nu\beta}-p^{2}g_{\mu\beta}g_{\nu\alpha}-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)
+ΠY(p2)(gμαpνpβgνβpμpα+gμβpνpα+gναpμpβ),subscriptΠ𝑌superscript𝑝2subscript𝑔𝜇𝛼subscript𝑝𝜈subscript𝑝𝛽subscript𝑔𝜈𝛽subscript𝑝𝜇subscript𝑝𝛼subscript𝑔𝜇𝛽subscript𝑝𝜈subscript𝑝𝛼subscript𝑔𝜈𝛼subscript𝑝𝜇subscript𝑝𝛽\displaystyle+\Pi_{Y}(p^{2})\left(-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)\,,
Πμναβ;2(p)subscriptΠ𝜇𝜈𝛼𝛽2𝑝\displaystyle\Pi_{\mu\nu\alpha\beta;2}(p) =\displaystyle= λZ2MZ2p2(g~μαg~νβ+g~μβg~να2g~μνg~αβ3)+,superscriptsubscript𝜆𝑍2superscriptsubscript𝑀𝑍2superscript𝑝2subscript~𝑔𝜇𝛼subscript~𝑔𝜈𝛽subscript~𝑔𝜇𝛽subscript~𝑔𝜈𝛼2subscript~𝑔𝜇𝜈subscript~𝑔𝛼𝛽3\displaystyle\frac{\lambda_{Z}^{2}}{M_{Z}^{2}-p^{2}}\left(\frac{\widetilde{g}_{\mu\alpha}\widetilde{g}_{\nu\beta}+\widetilde{g}_{\mu\beta}\widetilde{g}_{\nu\alpha}}{2}-\frac{\widetilde{g}_{\mu\nu}\widetilde{g}_{\alpha\beta}}{3}\right)+\cdots\,\,, (7)
=\displaystyle= Π2(p2)(g~μαg~νβ+g~μβg~να2g~μνg~αβ3),subscriptΠ2superscript𝑝2subscript~𝑔𝜇𝛼subscript~𝑔𝜈𝛽subscript~𝑔𝜇𝛽subscript~𝑔𝜈𝛼2subscript~𝑔𝜇𝜈subscript~𝑔𝛼𝛽3\displaystyle\Pi_{2}(p^{2})\left(\frac{\widetilde{g}_{\mu\alpha}\widetilde{g}_{\nu\beta}+\widetilde{g}_{\mu\beta}\widetilde{g}_{\nu\alpha}}{2}-\frac{\widetilde{g}_{\mu\nu}\widetilde{g}_{\alpha\beta}}{3}\right)\,,

where g~μν=gμνpμpνp2subscript~𝑔𝜇𝜈subscript𝑔𝜇𝜈subscript𝑝𝜇subscript𝑝𝜈superscript𝑝2\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}, the pole residues λZsubscript𝜆𝑍\lambda_{Z} and λYsubscript𝜆𝑌\lambda_{Y} are defined by

0|J0(0)|Z0+(p)quantum-operator-product0subscript𝐽00subscript𝑍superscript0𝑝\displaystyle\langle 0|J_{0}(0)|Z_{0^{+}}(p)\rangle =\displaystyle= λZ,subscript𝜆𝑍\displaystyle\lambda_{Z}\,,
0|Jμν;1(0)|Z1+(p)quantum-operator-product0subscript𝐽𝜇𝜈10subscript𝑍superscript1𝑝\displaystyle\langle 0|J_{\mu\nu;1}(0)|Z_{1^{+}}(p)\rangle =\displaystyle= λZMZϵμναβεαpβ,subscript𝜆𝑍subscript𝑀𝑍subscriptitalic-ϵ𝜇𝜈𝛼𝛽superscript𝜀𝛼superscript𝑝𝛽\displaystyle\frac{\lambda_{Z}}{M_{Z}}\,\epsilon_{\mu\nu\alpha\beta}\,\varepsilon^{\alpha}p^{\beta}\,,
0|Jμν;1(0)|Y1(p)quantum-operator-product0subscript𝐽𝜇𝜈10subscript𝑌superscript1𝑝\displaystyle\langle 0|J_{\mu\nu;1}(0)|Y_{1^{-}}(p)\rangle =\displaystyle= λYMY(εμpνενpμ),subscript𝜆𝑌subscript𝑀𝑌subscript𝜀𝜇subscript𝑝𝜈subscript𝜀𝜈subscript𝑝𝜇\displaystyle\frac{\lambda_{Y}}{M_{Y}}\left(\varepsilon_{\mu}p_{\nu}-\varepsilon_{\nu}p_{\mu}\right)\,,
0|Jμν;2(0)|Z2+(p)quantum-operator-product0subscript𝐽𝜇𝜈20subscript𝑍superscript2𝑝\displaystyle\langle 0|J_{\mu\nu;2}(0)|Z_{2^{+}}(p)\rangle =\displaystyle= λZεμν,subscript𝜆𝑍subscript𝜀𝜇𝜈\displaystyle\lambda_{Z}\,\varepsilon_{\mu\nu}\,, (8)

the εμsubscript𝜀𝜇\varepsilon_{\mu} and εμνsubscript𝜀𝜇𝜈\varepsilon_{\mu\nu} are the polarization vectors of the spin J=1𝐽1J=1 and 222 tetraquark states, respectively. The summation of the polarization vectors εμsubscript𝜀𝜇\varepsilon_{\mu} and εμνsubscript𝜀𝜇𝜈\varepsilon_{\mu\nu} results in the following formula,

λεμ(λ,p)εν(λ,p)subscript𝜆subscriptsuperscript𝜀𝜇𝜆𝑝subscript𝜀𝜈𝜆𝑝\displaystyle\sum_{\lambda}\varepsilon^{*}_{\mu}(\lambda,p)\varepsilon_{\nu}(\lambda,p) =\displaystyle= gμν+pμpνp2,subscript𝑔𝜇𝜈subscript𝑝𝜇subscript𝑝𝜈superscript𝑝2\displaystyle-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\,,
λεαβ(λ,p)εμν(λ,p)subscript𝜆subscriptsuperscript𝜀𝛼𝛽𝜆𝑝subscript𝜀𝜇𝜈𝜆𝑝\displaystyle\sum_{\lambda}\varepsilon^{*}_{\alpha\beta}(\lambda,p)\varepsilon_{\mu\nu}(\lambda,p) =\displaystyle= g~αμg~βν+g~ανg~βμ2g~αβg~μν3.subscript~𝑔𝛼𝜇subscript~𝑔𝛽𝜈subscript~𝑔𝛼𝜈subscript~𝑔𝛽𝜇2subscript~𝑔𝛼𝛽subscript~𝑔𝜇𝜈3\displaystyle\frac{\widetilde{g}_{\alpha\mu}\widetilde{g}_{\beta\nu}+\widetilde{g}_{\alpha\nu}\widetilde{g}_{\beta\mu}}{2}-\frac{\widetilde{g}_{\alpha\beta}\widetilde{g}_{\mu\nu}}{3}\,. (9)

The components Π0(p2)subscriptΠ0superscript𝑝2\Pi_{0}(p^{2}), ΠZ(p2)subscriptΠ𝑍superscript𝑝2\Pi_{Z}(p^{2}), ΠY(p2)subscriptΠ𝑌superscript𝑝2\Pi_{Y}(p^{2}) and Π2(p2)subscriptΠ2superscript𝑝2\Pi_{2}(p^{2}) receive contributions of the hadronic states with the spin-parity JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}, 1+superscript11^{+}, 1superscript11^{-} and 2+superscript22^{+}, respectively.

Now we project out the components ΠZ(p2)subscriptΠ𝑍superscript𝑝2\Pi_{Z}(p^{2}) and ΠY(p2)subscriptΠ𝑌superscript𝑝2\Pi_{Y}(p^{2}) by introducing the operators PZμναβsuperscriptsubscript𝑃𝑍𝜇𝜈𝛼𝛽P_{Z}^{\mu\nu\alpha\beta} and PYμναβsuperscriptsubscript𝑃𝑌𝜇𝜈𝛼𝛽P_{Y}^{\mu\nu\alpha\beta},

Π1;A(p2)subscriptΠ1𝐴superscript𝑝2\displaystyle\Pi_{1;A}(p^{2}) =\displaystyle= p2ΠZ(p2)=PZμναβΠμναβ;1(p),superscript𝑝2subscriptΠ𝑍superscript𝑝2superscriptsubscript𝑃𝑍𝜇𝜈𝛼𝛽subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle p^{2}\Pi_{Z}(p^{2})=P_{Z}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta;1}(p)\,,
Π1;V(p2)subscriptΠ1𝑉superscript𝑝2\displaystyle\Pi_{1;V}(p^{2}) =\displaystyle= p2ΠY(p2)=PYμναβΠμναβ;1(p),superscript𝑝2subscriptΠ𝑌superscript𝑝2superscriptsubscript𝑃𝑌𝜇𝜈𝛼𝛽subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle p^{2}\Pi_{Y}(p^{2})=P_{Y}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta;1}(p)\,, (10)

where

PZμναβsuperscriptsubscript𝑃𝑍𝜇𝜈𝛼𝛽\displaystyle P_{Z}^{\mu\nu\alpha\beta} =\displaystyle= 16(gμαpμpαp2)(gνβpνpβp2),16superscript𝑔𝜇𝛼superscript𝑝𝜇superscript𝑝𝛼superscript𝑝2superscript𝑔𝜈𝛽superscript𝑝𝜈superscript𝑝𝛽superscript𝑝2\displaystyle\frac{1}{6}\left(g^{\mu\alpha}-\frac{p^{\mu}p^{\alpha}}{p^{2}}\right)\left(g^{\nu\beta}-\frac{p^{\nu}p^{\beta}}{p^{2}}\right)\,,
PYμναβsuperscriptsubscript𝑃𝑌𝜇𝜈𝛼𝛽\displaystyle P_{Y}^{\mu\nu\alpha\beta} =\displaystyle= 16(gμαpμpαp2)(gνβpνpβp2)16gμαgνβ.16superscript𝑔𝜇𝛼superscript𝑝𝜇superscript𝑝𝛼superscript𝑝2superscript𝑔𝜈𝛽superscript𝑝𝜈superscript𝑝𝛽superscript𝑝216superscript𝑔𝜇𝛼superscript𝑔𝜈𝛽\displaystyle\frac{1}{6}\left(g^{\mu\alpha}-\frac{p^{\mu}p^{\alpha}}{p^{2}}\right)\left(g^{\nu\beta}-\frac{p^{\nu}p^{\beta}}{p^{2}}\right)-\frac{1}{6}g^{\mu\alpha}g^{\nu\beta}\,. (11)

In this article, we carry out the operator product expansion for the correlation functions Π0(p)subscriptΠ0𝑝\Pi_{0}(p), Πμναβ;1(p)subscriptΠ𝜇𝜈𝛼𝛽1𝑝\Pi_{\mu\nu\alpha\beta;1}(p) and Πμναβ;2(p)subscriptΠ𝜇𝜈𝛼𝛽2𝑝\Pi_{\mu\nu\alpha\beta;2}(p) to the vacuum condensates up to dimension-10, and take into account the vacuum condensates which are vacuum expectations of the operators of the orders 𝒪(αsk)𝒪superscriptsubscript𝛼𝑠𝑘\mathcal{O}(\alpha_{s}^{k}) with k1𝑘1k\leq 1 in a consistent way [18, 19, 20, 21, 22], then we project out the components

Π1;A(p2)subscriptΠ1𝐴superscript𝑝2\displaystyle\Pi_{1;A}(p^{2}) =\displaystyle= PZμναβΠμναβ;1(p),superscriptsubscript𝑃𝑍𝜇𝜈𝛼𝛽subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle P_{Z}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta;1}(p)\,,
Π1;V(p2)subscriptΠ1𝑉superscript𝑝2\displaystyle\Pi_{1;V}(p^{2}) =\displaystyle= PYμναβΠμναβ;1(p),superscriptsubscript𝑃𝑌𝜇𝜈𝛼𝛽subscriptΠ𝜇𝜈𝛼𝛽1𝑝\displaystyle P_{Y}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta;1}(p)\,, (12)

on the QCD side, and obtain the QCD spectral densities through dispersion relation,

ρ0(s)subscript𝜌0𝑠\displaystyle\rho_{0}(s) =\displaystyle= ImΠ0(s)π,ImsubscriptΠ0𝑠𝜋\displaystyle\frac{{\rm Im}\Pi_{0}(s)}{\pi}\,,
ρ1;A(s)subscript𝜌1𝐴𝑠\displaystyle\rho_{1;A}(s) =\displaystyle= ImΠ1;A(s)π,ImsubscriptΠ1𝐴𝑠𝜋\displaystyle\frac{{\rm Im}\Pi_{1;A}(s)}{\pi}\,,
ρ1;V(s)subscript𝜌1𝑉𝑠\displaystyle\rho_{1;V}(s) =\displaystyle= ImΠ1;V(s)π,ImsubscriptΠ1𝑉𝑠𝜋\displaystyle\frac{{\rm Im}\Pi_{1;V}(s)}{\pi}\,,
ρ2(s)subscript𝜌2𝑠\displaystyle\rho_{2}(s) =\displaystyle= ImΠ2(s)π,ImsubscriptΠ2𝑠𝜋\displaystyle\frac{{\rm Im}\Pi_{2}(s)}{\pi}\,, (13)

where ρ0(s)=ρu¯d¯;0(s)subscript𝜌0𝑠subscript𝜌¯𝑢¯𝑑0𝑠\rho_{0}(s)=\rho_{\bar{u}\bar{d};0}(s), ρu¯s¯;0(s)subscript𝜌¯𝑢¯𝑠0𝑠\rho_{\bar{u}\bar{s};0}(s), ρs¯s¯;0(s)subscript𝜌¯𝑠¯𝑠0𝑠\rho_{\bar{s}\bar{s};0}(s), ρ1;A(s)=ρu¯d¯;1;A(s)subscript𝜌1𝐴𝑠subscript𝜌¯𝑢¯𝑑1𝐴𝑠\rho_{1;A}(s)=\rho_{\bar{u}\bar{d};1;A}(s), ρu¯s¯;1;A(s)subscript𝜌¯𝑢¯𝑠1𝐴𝑠\rho_{\bar{u}\bar{s};1;A}(s), ρs¯s¯;1;A(s)subscript𝜌¯𝑠¯𝑠1𝐴𝑠\rho_{\bar{s}\bar{s};1;A}(s), ρ1;V(s)=ρu¯d¯;1;V(s)subscript𝜌1𝑉𝑠subscript𝜌¯𝑢¯𝑑1𝑉𝑠\rho_{1;V}(s)=\rho_{\bar{u}\bar{d};1;V}(s), ρu¯s¯;1;V(s)subscript𝜌¯𝑢¯𝑠1𝑉𝑠\rho_{\bar{u}\bar{s};1;V}(s), ρs¯s¯;1;V(s)subscript𝜌¯𝑠¯𝑠1𝑉𝑠\rho_{\bar{s}\bar{s};1;V}(s), ρ2(s)=ρu¯d¯;2(s)subscript𝜌2𝑠subscript𝜌¯𝑢¯𝑑2𝑠\rho_{2}(s)=\rho_{\bar{u}\bar{d};2}(s), ρu¯s¯;2(s)subscript𝜌¯𝑢¯𝑠2𝑠\rho_{\bar{u}\bar{s};2}(s), ρs¯s¯;2(s)subscript𝜌¯𝑠¯𝑠2𝑠\rho_{\bar{s}\bar{s};2}(s). The explicit expressions of the QCD spectral densities are given in the Appendix.

Once the analytical expressions of the QCD spectral densities ρ0(s)subscript𝜌0𝑠\rho_{0}(s), ρ1;A(s)subscript𝜌1𝐴𝑠\rho_{1;A}(s), ρ1;V(s)subscript𝜌1𝑉𝑠\rho_{1;V}(s), ρ2(s)subscript𝜌2𝑠\rho_{2}(s) are obtained, we can take the quark-hadron duality below the continuum thresholds s0subscript𝑠0s_{0} and perform Borel transform with respect to the variable P2=p2superscript𝑃2superscript𝑝2P^{2}=-p^{2} to obtain the QCD sum rules,

λZ/Y2exp(MZ/Y2T2)subscriptsuperscript𝜆2𝑍𝑌subscriptsuperscript𝑀2𝑍𝑌superscript𝑇2\displaystyle\lambda^{2}_{Z/Y}\,\exp\left(-\frac{M^{2}_{Z/Y}}{T^{2}}\right) =\displaystyle= 4mc2s0𝑑sρ(s)exp(sT2),superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠𝜌𝑠𝑠superscript𝑇2\displaystyle\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho(s)\,\exp\left(-\frac{s}{T^{2}}\right)\,, (14)

where ρ(s)=ρ0(s)𝜌𝑠subscript𝜌0𝑠\rho(s)=\rho_{0}(s), ρ1;A(s)subscript𝜌1𝐴𝑠\rho_{1;A}(s), ρ1;V(s)subscript𝜌1𝑉𝑠\rho_{1;V}(s), ρ2(s)subscript𝜌2𝑠\rho_{2}(s).

We derive Eq.(14) with respect to τ=1T2𝜏1superscript𝑇2\tau=\frac{1}{T^{2}}, then eliminate the pole residues λZ/Ysubscript𝜆𝑍𝑌\lambda_{Z/Y} to obtain the QCD sum rules for the masses of the doubly charmed tetraquark states,

MZ/Y2subscriptsuperscript𝑀2𝑍𝑌\displaystyle M^{2}_{Z/Y} =\displaystyle= ddτ4mc2s0𝑑sρ(s)eτs4mc2s0𝑑sρ(s)eτs.𝑑𝑑𝜏superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠𝜌𝑠superscript𝑒𝜏𝑠superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠𝜌𝑠superscript𝑒𝜏𝑠\displaystyle\frac{-\frac{d}{d\tau}\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho(s)\,e^{-\tau s}}{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho(s)\,e^{-\tau s}}\,. (15)

3 Numerical results and discussions

We take the standard values of the vacuum condensates q¯q=(0.24±0.01GeV)3delimited-⟨⟩¯𝑞𝑞superscriptplus-or-minus0.240.01GeV3\langle\bar{q}q\rangle=-(0.24\pm 0.01\,\rm{GeV})^{3}, q¯gsσGq=m02q¯qdelimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞superscriptsubscript𝑚02delimited-⟨⟩¯𝑞𝑞\langle\bar{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle, m02=(0.8±0.1)GeV2superscriptsubscript𝑚02plus-or-minus0.80.1superscriptGeV2m_{0}^{2}=(0.8\pm 0.1)\,\rm{GeV}^{2}, s¯s=(0.8±0.1)q¯qdelimited-⟨⟩¯𝑠𝑠plus-or-minus0.80.1delimited-⟨⟩¯𝑞𝑞\langle\bar{s}s\rangle=(0.8\pm 0.1)\langle\bar{q}q\rangle, s¯gsσGs=m02s¯sdelimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠superscriptsubscript𝑚02delimited-⟨⟩¯𝑠𝑠\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle, αsGGπ=(0.33GeV)4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscript0.33GeV4\langle\frac{\alpha_{s}GG}{\pi}\rangle=(0.33\,\rm{GeV})^{4} at the energy scale μ=1GeV𝜇1GeV\mu=1\,\rm{GeV} [6, 7, 23], and choose the MS¯¯𝑀𝑆\overline{MS} masses mc(mc)=(1.275±0.025)GeVsubscript𝑚𝑐subscript𝑚𝑐plus-or-minus1.2750.025GeVm_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}, ms(μ=2GeV)=(0.095±0.005)GeVsubscript𝑚𝑠𝜇2GeVplus-or-minus0.0950.005GeVm_{s}(\mu=2\,\rm{GeV})=(0.095\pm 0.005)\,\rm{GeV} from the Particle Data Group [24]. Moreover, we take into account the energy-scale dependence of the input parameters on the QCD side,

q¯q(μ)delimited-⟨⟩¯𝑞𝑞𝜇\displaystyle\langle\bar{q}q\rangle(\mu) =\displaystyle= q¯q(Q)[αs(Q)αs(μ)]49,delimited-⟨⟩¯𝑞𝑞𝑄superscriptdelimited-[]subscript𝛼𝑠𝑄subscript𝛼𝑠𝜇49\displaystyle\langle\bar{q}q\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{4}{9}}\,,
s¯s(μ)delimited-⟨⟩¯𝑠𝑠𝜇\displaystyle\langle\bar{s}s\rangle(\mu) =\displaystyle= s¯s(Q)[αs(Q)αs(μ)]49,delimited-⟨⟩¯𝑠𝑠𝑄superscriptdelimited-[]subscript𝛼𝑠𝑄subscript𝛼𝑠𝜇49\displaystyle\langle\bar{s}s\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{4}{9}}\,,
q¯gsσGq(μ)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞𝜇\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu) =\displaystyle= q¯gsσGq(Q)[αs(Q)αs(μ)]227,delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞𝑄superscriptdelimited-[]subscript𝛼𝑠𝑄subscript𝛼𝑠𝜇227\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{2}{27}}\,,
s¯gsσGs(μ)delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠𝜇\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle(\mu) =\displaystyle= s¯gsσGs(Q)[αs(Q)αs(μ)]227,delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠𝑄superscriptdelimited-[]subscript𝛼𝑠𝑄subscript𝛼𝑠𝜇227\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle(Q)\left[\frac{\alpha_{s}(Q)}{\alpha_{s}(\mu)}\right]^{\frac{2}{27}}\,,
mc(μ)subscript𝑚𝑐𝜇\displaystyle m_{c}(\mu) =\displaystyle= mc(mc)[αs(μ)αs(mc)]1225,subscript𝑚𝑐subscript𝑚𝑐superscriptdelimited-[]subscript𝛼𝑠𝜇subscript𝛼𝑠subscript𝑚𝑐1225\displaystyle m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{25}}\,,
ms(μ)subscript𝑚𝑠𝜇\displaystyle m_{s}(\mu) =\displaystyle= ms(2GeV)[αs(μ)αs(2GeV)]49,subscript𝑚𝑠2GeVsuperscriptdelimited-[]subscript𝛼𝑠𝜇subscript𝛼𝑠2GeV49\displaystyle m_{s}({\rm 2GeV})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\rm 2GeV})}\right]^{\frac{4}{9}}\,,
αs(μ)subscript𝛼𝑠𝜇\displaystyle\alpha_{s}(\mu) =\displaystyle= 1b0t[1b1b02logtt+b12(log2tlogt1)+b0b2b04t2],1subscript𝑏0𝑡delimited-[]1subscript𝑏1superscriptsubscript𝑏02𝑡𝑡superscriptsubscript𝑏12superscript2𝑡𝑡1subscript𝑏0subscript𝑏2superscriptsubscript𝑏04superscript𝑡2\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,, (16)

where t=logμ2Λ2𝑡superscript𝜇2superscriptΛ2t=\log\frac{\mu^{2}}{\Lambda^{2}}, b0=332nf12πsubscript𝑏0332subscript𝑛𝑓12𝜋b_{0}=\frac{33-2n_{f}}{12\pi}, b1=15319nf24π2subscript𝑏115319subscript𝑛𝑓24superscript𝜋2b_{1}=\frac{153-19n_{f}}{24\pi^{2}}, b2=285750339nf+32527nf2128π3subscript𝑏2285750339subscript𝑛𝑓32527superscriptsubscript𝑛𝑓2128superscript𝜋3b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}, Λ=213MeVΛ213MeV\Lambda=213\,\rm{MeV}, 296MeV296MeV296\,\rm{MeV} and 339MeV339MeV339\,\rm{MeV} for the flavors nf=5subscript𝑛𝑓5n_{f}=5, 444 and 333, respectively [24], and evolve all the input parameters to the optimal energy scales μ𝜇\mu to extract the masses of the doubly charmed tetraquark states Z𝑍Z and Y𝑌Y.

In the article, we study the doubly charmed tetraquark states, the two charm quarks form an axialvector doubly charmed diquark state in color antitriplet, the axialvector doubly charmed diquark state serves as a static well potential and combines with an axialvector light antidiquark state in color triplet to form a compact tetraquark state. While in the hidden-charm tetraquark states, the charm quark c𝑐c serves as a static well potential and combines with the light quark q𝑞q to form a charmed diquark in color antitriplet, the charm antiquark c¯¯𝑐\bar{c} serves as another static well potential and combines with the light antiquark q¯superscript¯𝑞\bar{q}^{\prime} to form a charmed antidiquark in color triplet, then the charmed diquark and charmed antidiquark combine together to form a hidden-charm tetraquark state. The quark structures of the doubly charmed tetraquark states and hidden-charm tetraquark states are quite different.

In Refs.[18, 19, 20, 21, 22], we study the acceptable energy scales of the QCD spectral densities for the hidden-charm (hidden-bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest an energy scale formula μ=MX/Y/Z2(2𝕄Q)2𝜇subscriptsuperscript𝑀2𝑋𝑌𝑍superscript2subscript𝕄𝑄2\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{Q})^{2}} to determine the optimal energy scales. The energy scale formula also works well in studying the hidden-charm pentaquark states [25]. The updated values of the effective heavy quark masses are 𝕄c=1.82GeVsubscript𝕄𝑐1.82GeV{\mathbb{M}}_{c}=1.82\,\rm{GeV} and 𝕄b=5.17GeVsubscript𝕄𝑏5.17GeV{\mathbb{M}}_{b}=5.17\,\rm{GeV} [26]. It is not necessary for the effective charm quark mass 𝕄csubscript𝕄𝑐{\mathbb{M}}_{c} in the doubly charmed tetraquark states to have the same value as the one in the hidden-charm tetraquark states. In calculations, we observe that if we choose a slightly different value 𝕄c=1.84GeVsubscript𝕄𝑐1.84GeV{\mathbb{M}}_{c}=1.84\,\rm{GeV}, the criteria of the QCD sum rules can be satisfied more easily. We obtain the energy scale formula by setting the energy scale μ=V𝜇𝑉\mu=V, the virtuality V𝑉V (or bound energy not as robust) is defined by V=MX/Y/Z2(2𝕄c)2𝑉subscriptsuperscript𝑀2𝑋𝑌𝑍superscript2subscript𝕄𝑐2V=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}} [19, 20]. In this article, we take into account the SU(3)𝑆𝑈3SU(3) breaking effect ms(μ)subscript𝑚𝑠𝜇m_{s}(\mu) by subtracting the ms(μ)subscript𝑚𝑠𝜇m_{s}(\mu) from the virtuality V𝑉V, μk=Vk=MX/Y/Z2(2𝕄c)2kms(μk)subscript𝜇𝑘subscript𝑉𝑘subscriptsuperscript𝑀2𝑋𝑌𝑍superscript2subscript𝕄𝑐2𝑘subscript𝑚𝑠subscript𝜇𝑘\mu_{k}=V_{k}=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}}-k\,m_{s}(\mu_{k}), where the numbers of the strange antiquark s¯¯𝑠\bar{s} in the doubly charmed tetraquark states are k=0,1,2𝑘012k=0,1,2.

In this article, we take the continuum threshold parameters as s0=MZ/Y+(0.40.7)GeVsubscript𝑠0subscript𝑀𝑍𝑌similar-to0.40.7GeV\sqrt{s_{0}}=M_{Z/Y}+(0.4\sim 0.7)\,\rm{GeV}, and vary the parameters s0subscript𝑠0\sqrt{s_{0}} to obtain the optimal Borel parameters T2superscript𝑇2T^{2} to satisfy the following four criteria:

𝟏.1\bf 1. Pole dominance on the phenomenological side;

𝟐.2\bf 2. Convergence of the operator product expansion;

𝟑.3\bf 3. Appearance of the Borel platforms;

𝟒.4\bf 4. Satisfying the energy scale formula.

The resulting Borel parameters or Borel windows T2superscript𝑇2T^{2}, continuum threshold parameters s0subscript𝑠0s_{0}, optimal energy scales of the QCD spectral densities, pole contributions of the ground states are shown explicitly in Table 1. From Table 1, we can see that the pole dominance can be well satisfied. The pole contributions PCPC\rm{PC} are defined by

PCPC\displaystyle{\rm PC} =\displaystyle= 4mc2s0𝑑sρ(s)exp(sT2)4mc2𝑑sρ(s)exp(sT2),superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠𝜌𝑠𝑠superscript𝑇2superscriptsubscript4superscriptsubscript𝑚𝑐2differential-d𝑠𝜌𝑠𝑠superscript𝑇2\displaystyle\frac{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho(s)\,\exp\left(-\frac{s}{T^{2}}\right)}{\int_{4m_{c}^{2}}^{\infty}ds\,\rho(s)\,\exp\left(-\frac{s}{T^{2}}\right)}\,, (17)

which decrease monotonously and quickly with increase of the Borel parameter T2superscript𝑇2T^{2}, as the continuum contributions are depressed by the factor exp(sT2)𝑠superscript𝑇2\exp\left(-\frac{s}{T^{2}}\right), large Borel parameter T2superscript𝑇2T^{2} enhances the continuum contributions, the largest power of the QCD spectral densities ρ(s)s4proportional-to𝜌𝑠superscript𝑠4\rho(s)\propto s^{4}, the convergent behaviors of the operator product expansion are not very good for the tetraquark states and molecular states. Furthermore, the pole contributions increase monotonously with increase of the threshold parameters s0subscript𝑠0s_{0}, the uncertainties of the threshold parameters δs0=±0.1GeV𝛿subscript𝑠0plus-or-minus0.1GeV\delta\sqrt{s_{0}}=\pm 0.1\,\rm{GeV} also lead to rather large variations of the pole contributions. So in the small Borel window Tmax2Tmin2=0.4GeV2subscriptsuperscript𝑇2𝑚𝑎𝑥subscriptsuperscript𝑇2𝑚𝑖𝑛0.4superscriptGeV2T^{2}_{max}-T^{2}_{min}=0.4\,\rm{GeV}^{2} for the JP=(0/1/2)+superscript𝐽𝑃superscript012J^{P}=(0/1/2)^{+} tetraquark states, the pole contributions vary in a rather large range, about (4060)%percent4060(40-60)\%. Although the pole contributions have rather large uncertainties, PC=(50±10)%PCpercentplus-or-minus5010{\rm PC}=(50\pm 10)\% for the JP=(0/1/2)+superscript𝐽𝑃superscript012J^{P}=(0/1/2)^{+} tetraquark states and PC=(60±10)%PCpercentplus-or-minus6010{\rm PC}=(60\pm 10)\% for the JP=1superscript𝐽𝑃superscript1J^{P}=1^{-} tetraquark states, the pole dominance can be well satisfied, the predictions are reliable. On the other hand, if we choose larger energy scales μ𝜇\mu, the pole contributions are enhanced, the pole contributions are less sensitive to the Borel parameter T2superscript𝑇2T^{2}, however, we should determine the energy scales of the QCD spectral densities in a consistent way by using the energy scale formula.

In Fig.1, we plot the absolute contributions of the vacuum condensates |D(n)|𝐷𝑛|D(n)| in the operator product expansion for the central values of the input parameters,

D(n)𝐷𝑛\displaystyle D(n) =\displaystyle= 4mc2s0𝑑sρn(s)exp(sT2)4mc2s0𝑑sρ(s)exp(sT2),superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠subscript𝜌𝑛𝑠𝑠superscript𝑇2superscriptsubscript4superscriptsubscript𝑚𝑐2subscript𝑠0differential-d𝑠𝜌𝑠𝑠superscript𝑇2\displaystyle\frac{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{n}(s)\,\exp\left(-\frac{s}{T^{2}}\right)}{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho(s)\,\exp\left(-\frac{s}{T^{2}}\right)}\,, (18)

where the ρn(s)subscript𝜌𝑛𝑠\rho_{n}(s) are the QCD spectral densities for the vacuum condensates of dimension n𝑛n. From the figure, we can see that the dominant contributions come from the perturbative terms (or D(0)𝐷0D(0)) for the 1superscript11^{-} tetraquark states, the operator product expansion is well convergent, while in the case of the 0+superscript00^{+}, 1+superscript11^{+} and 2+superscript22^{+} tetraquark states, the contributions of the vacuum condensates of dimension n=6𝑛6n=6 are very large, but the contributions of the vacuum condensates of dimensions 6, 8, 1068106,\,8,\,10 have the hierarchy |D(6)||D(8)||D(10)|much-greater-than𝐷6𝐷8much-greater-than𝐷10|D(6)|\gg|D(8)|\gg|D(10)|, the operator product expansion is also convergent.

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Figure 1: The absolute contributions of the vacuum condensates of dimension n𝑛n for central values of the input parameters, where the (I), (II), (III) and (IV) denote the tetraquark states with JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}, 1+superscript11^{+}, 2+superscript22^{+} and 1superscript11^{-} respectively, the A𝐴A, B𝐵B and C𝐶C denote the quark constituents ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d}, ccu¯s¯𝑐𝑐¯𝑢¯𝑠cc\bar{u}\bar{s} and ccs¯s¯𝑐𝑐¯𝑠¯𝑠cc\bar{s}\bar{s} respectively.

We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the doubly charmed tetraquark states Z𝑍Z and Y𝑌Y, which are shown explicitly in Table 1 and Figs.2-5. In Figs.2-5, we plot the masses and pole residues of the doubly charmed tetraquark states in much large ranges than the Borel windows. From Figs.2-5, we can see that there appear platforms in the Borel windows shown in Table 1. Furthermore, from Table 1, we can see that the energy scale formula μk=MX/Y/Z2(2𝕄c)2kms(μk)subscript𝜇𝑘subscriptsuperscript𝑀2𝑋𝑌𝑍superscript2subscript𝕄𝑐2𝑘subscript𝑚𝑠subscript𝜇𝑘\mu_{k}=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c})^{2}}-k\,m_{s}(\mu_{k}) with k=0,1,2𝑘012k=0,1,2 is also satisfied. Now the four criteria are all satisfied, we expect to make reliable predictions.

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Figure 2: The masses with variations of the Borel parameters, where the A𝐴A, B𝐵B, C𝐶C, D𝐷D, E𝐸E and F𝐹F denote the tetraquark states ccu¯d¯(0+)𝑐𝑐¯𝑢¯𝑑superscript0cc\bar{u}\bar{d}\,(0^{+}), ccu¯s¯(0+)𝑐𝑐¯𝑢¯𝑠superscript0cc\bar{u}\bar{s}\,(0^{+}), ccs¯s¯(0+)𝑐𝑐¯𝑠¯𝑠superscript0cc\bar{s}\bar{s}\,(0^{+}), ccu¯d¯(1+)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}\,(1^{+}), ccu¯s¯(1+)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}\,(1^{+}) and ccs¯s¯(1+)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}\,(1^{+}), respectively.
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Figure 3: The masses with variations of the Borel parameters, where the A𝐴A, B𝐵B, C𝐶C, D𝐷D, E𝐸E and F𝐹F denote the tetraquark states ccu¯d¯(2+)𝑐𝑐¯𝑢¯𝑑superscript2cc\bar{u}\bar{d}\,(2^{+}), ccu¯s¯(2+)𝑐𝑐¯𝑢¯𝑠superscript2cc\bar{u}\bar{s}\,(2^{+}), ccs¯s¯(2+)𝑐𝑐¯𝑠¯𝑠superscript2cc\bar{s}\bar{s}\,(2^{+}), ccu¯d¯(1)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}\,(1^{-}), ccu¯s¯(1)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}\,(1^{-}) and ccs¯s¯(1)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}\,(1^{-}), respectively.
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Figure 4: The pole residues with variations of the Borel parameters, where the A𝐴A, B𝐵B, C𝐶C, D𝐷D, E𝐸E and F𝐹F denote the tetraquark states ccu¯d¯(0+)𝑐𝑐¯𝑢¯𝑑superscript0cc\bar{u}\bar{d}\,(0^{+}), ccu¯s¯(0+)𝑐𝑐¯𝑢¯𝑠superscript0cc\bar{u}\bar{s}\,(0^{+}), ccs¯s¯(0+)𝑐𝑐¯𝑠¯𝑠superscript0cc\bar{s}\bar{s}\,(0^{+}), ccu¯d¯(1+)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}\,(1^{+}), ccu¯s¯(1+)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}\,(1^{+}) and ccs¯s¯(1+)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}\,(1^{+}), respectively.
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Figure 5: The pole residues with variations of the Borel parameters, where the A𝐴A, B𝐵B, C𝐶C, D𝐷D, E𝐸E and F𝐹F denote the tetraquark states ccu¯d¯(2+)𝑐𝑐¯𝑢¯𝑑superscript2cc\bar{u}\bar{d}\,(2^{+}), ccu¯s¯(2+)𝑐𝑐¯𝑢¯𝑠superscript2cc\bar{u}\bar{s}\,(2^{+}), ccs¯s¯(2+)𝑐𝑐¯𝑠¯𝑠superscript2cc\bar{s}\bar{s}\,(2^{+}), ccu¯d¯(1)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}\,(1^{-}), ccu¯s¯(1)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}\,(1^{-}) and ccs¯s¯(1)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}\,(1^{-}), respectively.

In Ref.[20], we tentatively assign the Zc(4020/4025)subscript𝑍𝑐40204025Z_{c}(4020/4025) to be the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type hidden-charm axialvector tetraquark state, and choose the current,

Jμν;cc¯(x)subscript𝐽𝜇𝜈𝑐¯𝑐𝑥\displaystyle J_{\mu\nu;c\bar{c}}(x) =\displaystyle= εijkεimn{ujT(x)Cγμck(x)d¯m(x)γνCc¯nT(x)ujT(x)Cγνck(x)d¯m(x)γμCc¯nT(x)},superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛subscriptsuperscript𝑢𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑑𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑐𝑇𝑛𝑥subscriptsuperscript𝑢𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑑𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑐𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\left\{u^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\bar{d}_{m}(x)\gamma_{\nu}C\bar{c}^{T}_{n}(x)-u^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\bar{d}_{m}(x)\gamma_{\mu}C\bar{c}^{T}_{n}(x)\right\}\,,

to study it with the QCD sum rules. In Ref.[17], we choose the axialvector current Jμ;cc(x)subscript𝐽𝜇𝑐𝑐𝑥J_{\mu;cc}(x),

Jμ;cc(x)subscript𝐽𝜇𝑐𝑐𝑥\displaystyle J_{\mu;cc}(x) =\displaystyle= εijkεimncjT(x)Cγμck(x)u¯m(x)γ5Cd¯nT(x),superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾5𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{5}C\bar{d}^{T}_{n}(x)\,, (20)

to study the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type doubly charmed tetraquark state with the QCD sum rules. In this article, we choose the axialvector current Jμν;u¯d¯;1(x)subscript𝐽𝜇𝜈¯𝑢¯𝑑1𝑥J_{\mu\nu;\bar{u}\bar{d};1}(x),

Jμν;u¯d¯;1(x)subscript𝐽𝜇𝜈¯𝑢¯𝑑1𝑥\displaystyle J_{\mu\nu;\bar{u}\bar{d};1}(x) =\displaystyle= εijkεimn[cjT(x)Cγμck(x)u¯m(x)γνCd¯nT(x)cjT(x)Cγνck(x)u¯m(x)γμCd¯nT(x)],superscript𝜀𝑖𝑗𝑘superscript𝜀𝑖𝑚𝑛delimited-[]subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜇subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜈𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥subscriptsuperscript𝑐𝑇𝑗𝑥𝐶subscript𝛾𝜈subscript𝑐𝑘𝑥subscript¯𝑢𝑚𝑥subscript𝛾𝜇𝐶subscriptsuperscript¯𝑑𝑇𝑛𝑥\displaystyle\varepsilon^{ijk}\varepsilon^{imn}\,\left[c^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\nu}C\bar{d}^{T}_{n}(x)-c^{T}_{j}(x)C\gamma_{\nu}c_{k}(x)\,\bar{u}_{m}(x)\gamma_{\mu}C\bar{d}^{T}_{n}(x)\right]\,,

to study the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type doubly charmed tetraquark state.

In Fig.6, we plot the masses of the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type axialvector tetraquark state cc¯ud¯𝑐¯𝑐𝑢¯𝑑c\bar{c}u\bar{d}, Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type axialvector tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d} and CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type axialvector tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d} with variations of the Borel parameter T2superscript𝑇2T^{2} for the energy scale μ=1.3GeV𝜇1.3GeV\mu=1.3\,\rm{GeV} and continuum threshold parameter s0=4.45GeVsubscript𝑠04.45GeV\sqrt{s_{0}}=4.45\,\rm{GeV}. From the figure, we can see that the mass of the axialvector hidden-charm tetraquark state is 0.1GeV0.1GeV0.1\,\rm{GeV} larger than the ones of the corresponding axialvector doubly charmed tetraquark states, while the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type and CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type axialvector tetraquark states ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d} have almost degenerate masses. In Ref.[20], we observe that the calculations based on the QCD sum rules support that the Zc(4020/4025)subscript𝑍𝑐40204025Z_{c}(4020/4025) can be assigned to be the axialvector hidden-charm tetraquark state. So the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type and CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type axialvector tetraquark states ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d} have the masses about 3.9GeV3.9GeV3.9\,\rm{GeV}, the present predictions are reasonable. In Ref.[17] and present work, we observe that we can choose a universal effective c𝑐c-quark mass 𝕄c=1.84GeVsubscript𝕄𝑐1.84GeV{\mathbb{M}}_{c}=1.84\,\rm{GeV} to determine the energy scales of the QCD spectral densities in a consistent way, which leads to the energy scale μ=1.3GeV𝜇1.3GeV\mu=1.3\,\rm{GeV} for the QCD spectral density of the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d}. If we choose a slightly different energy scale μ=1.4GeV𝜇1.4GeV\mu=1.4\,\rm{GeV} (which corresponds to a non-universal value 𝕄c=1.82GeVsubscript𝕄𝑐1.82GeV{\mathbb{M}}_{c}=1.82\,\rm{GeV}) and a slightly different threshold parameter, we can obtain the lowest mass 3.85±0.09GeVplus-or-minus3.850.09GeV3.85\pm 0.09\,\rm{GeV}, which is also shown in Table 1 in Ref.[17]. In this article, we prefer the universal effective c𝑐c-quark mass 𝕄c=1.84GeVsubscript𝕄𝑐1.84GeV{\mathbb{M}}_{c}=1.84\,\rm{GeV}.

The centroids of the masses of the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type tetraquark states are

MCγμγνC(ccu¯d¯)subscript𝑀tensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶𝑐𝑐¯𝑢¯𝑑\displaystyle M_{C\gamma_{\mu}\otimes\gamma_{\nu}C}(cc\bar{u}\bar{d}) =\displaystyle= Mccu¯d¯;0++3Mccu¯d¯;1++5Mccu¯d¯;2+9=3.92GeV,subscript𝑀𝑐𝑐¯𝑢¯𝑑superscript03subscript𝑀𝑐𝑐¯𝑢¯𝑑superscript15subscript𝑀𝑐𝑐¯𝑢¯𝑑superscript293.92GeV\displaystyle\frac{M_{cc\bar{u}\bar{d};0^{+}}+3M_{cc\bar{u}\bar{d};1^{+}}+5M_{cc\bar{u}\bar{d};2^{+}}}{9}=3.92\,\rm{GeV}\,,
MCγμγνC(ccu¯s¯)subscript𝑀tensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶𝑐𝑐¯𝑢¯𝑠\displaystyle M_{C\gamma_{\mu}\otimes\gamma_{\nu}C}(cc\bar{u}\bar{s}) =\displaystyle= Mccu¯s¯;0++3Mccu¯s¯;1++5Mccu¯s¯;2+9=3.99GeV,subscript𝑀𝑐𝑐¯𝑢¯𝑠superscript03subscript𝑀𝑐𝑐¯𝑢¯𝑠superscript15subscript𝑀𝑐𝑐¯𝑢¯𝑠superscript293.99GeV\displaystyle\frac{M_{cc\bar{u}\bar{s};0^{+}}+3M_{cc\bar{u}\bar{s};1^{+}}+5M_{cc\bar{u}\bar{s};2^{+}}}{9}=3.99\,\rm{GeV}\,,
MCγμγνC(ccs¯s¯)subscript𝑀tensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶𝑐𝑐¯𝑠¯𝑠\displaystyle M_{C\gamma_{\mu}\otimes\gamma_{\nu}C}(cc\bar{s}\bar{s}) =\displaystyle= Mccs¯s¯;0++3Mccs¯s¯;1++5Mccs¯s¯;2+9=4.04GeV,subscript𝑀𝑐𝑐¯𝑠¯𝑠superscript03subscript𝑀𝑐𝑐¯𝑠¯𝑠superscript15subscript𝑀𝑐𝑐¯𝑠¯𝑠superscript294.04GeV\displaystyle\frac{M_{cc\bar{s}\bar{s};0^{+}}+3M_{cc\bar{s}\bar{s};1^{+}}+5M_{cc\bar{s}\bar{s};2^{+}}}{9}=4.04\,\rm{GeV}\,, (22)

which are slightly larger than the centroids of the masses of the corresponding Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type tetraquark states,

MCγμγ5C(ccu¯d¯)subscript𝑀tensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶𝑐𝑐¯𝑢¯𝑑\displaystyle M_{C\gamma_{\mu}\otimes\gamma_{5}C}(cc\bar{u}\bar{d}) =\displaystyle= 3.90GeV,3.90GeV\displaystyle 3.90\,\rm{GeV}\,,
MCγμγ5C(ccu¯s¯)subscript𝑀tensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶𝑐𝑐¯𝑢¯𝑠\displaystyle M_{C\gamma_{\mu}\otimes\gamma_{5}C}(cc\bar{u}\bar{s}) =\displaystyle= 3.95GeV,3.95GeV\displaystyle 3.95\,\rm{GeV}\,, (23)

so the ground states are the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type tetraquark states, which is consistent with our naive expectation that the axialvector (anti)diquarks have larger masses than the corresponding scalar (anti)diquarks. The lowest centroids Mccu¯d¯;0+=3.87GeVsubscript𝑀𝑐𝑐¯𝑢¯𝑑superscript03.87GeVM_{cc\bar{u}\bar{d};0^{+}}=3.87\,\rm{GeV} and Mccu¯s¯;0+=3.94GeVsubscript𝑀𝑐𝑐¯𝑢¯𝑠superscript03.94GeVM_{cc\bar{u}\bar{s};0^{+}}=3.94\,\rm{GeV} originate from the spin splitting, in other words, the spin-spin interaction between the doubly heavy diquark and the light antidiquark. In fact, the predicted masses have uncertainties, the centroids of the masses are not the super values, all values within uncertainties make sense.

In Ref.[14], Eichten and Quigg obtain the masses M=4146MeV𝑀4146MeVM=4146\,\rm{MeV}, 4167MeV4167MeV4167\,\rm{MeV} and 4210MeV4210MeV4210\,\rm{MeV} for the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type axialvector tetraquark states ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d}, ccu¯s¯𝑐𝑐¯𝑢¯𝑠cc\bar{u}\bar{s} and ccs¯s¯𝑐𝑐¯𝑠¯𝑠cc\bar{s}\bar{s} respectively, which are about 0.200.25GeV0.200.25GeV0.20-0.25\,\rm{GeV} larger than the central values of the present predictions. For the Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type axialvector tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d}, Eichten and Quigg obtain the mass M=3978MeV𝑀3978MeVM=3978\,\rm{MeV} [14], which is 0.1GeV0.1GeV0.1\,\rm{GeV} larger than the value 3882MeV3882MeV3882\,\rm{MeV} obtained by Karliner and Rosner based on a simple potential quark model [9]. The present predictions are consistent with the value 3882MeV3882MeV3882\,\rm{MeV} obtained by Karliner and Rosner.

The doubly charmed tetraquark states with the JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}, 1+superscript11^{+} and 2+superscript22^{+} lie near the corresponding charmed meson pair thresholds, the decays to the charmed meson pairs are Okubo-Zweig-Iizuka super-allowed,

Zccu¯d¯;0+subscript𝑍𝑐𝑐¯𝑢¯𝑑superscript0\displaystyle Z_{cc\bar{u}\bar{d};0^{+}} \displaystyle\to D0D+,superscript𝐷0superscript𝐷\displaystyle D^{0}D^{+}\,,
Zccu¯s¯;0+subscript𝑍𝑐𝑐¯𝑢¯𝑠superscript0\displaystyle Z_{cc\bar{u}\bar{s};0^{+}} \displaystyle\to D0Ds+,superscript𝐷0superscriptsubscript𝐷𝑠\displaystyle D^{0}D_{s}^{+}\,,
Zccs¯s¯;0+subscript𝑍𝑐𝑐¯𝑠¯𝑠superscript0\displaystyle Z_{cc\bar{s}\bar{s};0^{+}} \displaystyle\to Ds+Ds+,subscriptsuperscript𝐷𝑠superscriptsubscript𝐷𝑠\displaystyle D^{+}_{s}D_{s}^{+}\,,
Zccu¯d¯;1+subscript𝑍𝑐𝑐¯𝑢¯𝑑superscript1\displaystyle Z_{cc\bar{u}\bar{d};1^{+}} \displaystyle\to D0D+,D+D0,superscript𝐷0superscript𝐷absentsuperscript𝐷superscript𝐷absent0\displaystyle D^{0}D^{*+}\,,\,\,D^{+}D^{*0}\,,
Zccu¯s¯;1+subscript𝑍𝑐𝑐¯𝑢¯𝑠superscript1\displaystyle Z_{cc\bar{u}\bar{s};1^{+}} \displaystyle\to D0Ds+,Ds+D0,superscript𝐷0superscriptsubscript𝐷𝑠absentsubscriptsuperscript𝐷𝑠superscript𝐷absent0\displaystyle D^{0}D_{s}^{*+}\,,\,\,D^{+}_{s}D^{*0}\,,
Zccs¯s¯;1+subscript𝑍𝑐𝑐¯𝑠¯𝑠superscript1\displaystyle Z_{cc\bar{s}\bar{s};1^{+}} \displaystyle\to Ds+Ds+,subscriptsuperscript𝐷𝑠superscriptsubscript𝐷𝑠absent\displaystyle D^{+}_{s}D_{s}^{*+}\,,
Zccu¯d¯;2+subscript𝑍𝑐𝑐¯𝑢¯𝑑superscript2\displaystyle Z_{cc\bar{u}\bar{d};2^{+}} \displaystyle\to D0D+,D0D+,superscript𝐷0superscript𝐷superscript𝐷absent0superscript𝐷absent\displaystyle D^{0}D^{+}\,,\,\,D^{*0}D^{*+}\,,
Zccu¯s¯;2+subscript𝑍𝑐𝑐¯𝑢¯𝑠superscript2\displaystyle Z_{cc\bar{u}\bar{s};2^{+}} \displaystyle\to D0Ds+,superscript𝐷0superscriptsubscript𝐷𝑠\displaystyle D^{0}D_{s}^{+}\,,
Zccs¯s¯;2+subscript𝑍𝑐𝑐¯𝑠¯𝑠superscript2\displaystyle Z_{cc\bar{s}\bar{s};2^{+}} \displaystyle\to Ds+Ds+,subscriptsuperscript𝐷𝑠superscriptsubscript𝐷𝑠\displaystyle D^{+}_{s}D_{s}^{+}\,, (24)

but the available phase spaces are very small, the decays are kinematically depressed, the doubly charmed tetraquark states with the JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}, 1+superscript11^{+} and 2+superscript22^{+} maybe have small widths. On the other hand, the doubly charmed tetraquark states with the JP=1superscript𝐽𝑃superscript1J^{P}=1^{-} lie above the corresponding charmed meson pair thresholds, the decays to the charmed meson pairs are Okubo-Zweig-Iizuka super-allowed,

Yccu¯d¯;1subscript𝑌𝑐𝑐¯𝑢¯𝑑superscript1\displaystyle Y_{cc\bar{u}\bar{d};1^{-}} \displaystyle\to D0D+,D0D+,D+D0,superscript𝐷0superscript𝐷superscript𝐷0superscript𝐷absentsuperscript𝐷superscript𝐷absent0\displaystyle D^{0}D^{+}\,,\,\,D^{0}D^{*+}\,,\,\,D^{+}D^{*0}\,,
Yccu¯s¯;1subscript𝑌𝑐𝑐¯𝑢¯𝑠superscript1\displaystyle Y_{cc\bar{u}\bar{s};1^{-}} \displaystyle\to D0Ds+,D0Ds+,Ds+D0,superscript𝐷0superscriptsubscript𝐷𝑠superscript𝐷0superscriptsubscript𝐷𝑠absentsubscriptsuperscript𝐷𝑠superscript𝐷absent0\displaystyle D^{0}D_{s}^{+}\,,\,\,D^{0}D_{s}^{*+}\,,\,\,D^{+}_{s}D^{*0}\,,
Yccs¯s¯;1subscript𝑌𝑐𝑐¯𝑠¯𝑠superscript1\displaystyle Y_{cc\bar{s}\bar{s};1^{-}} \displaystyle\to Ds+Ds+,Ds+Ds+,subscriptsuperscript𝐷𝑠superscriptsubscript𝐷𝑠subscriptsuperscript𝐷𝑠superscriptsubscript𝐷𝑠absent\displaystyle D^{+}_{s}D_{s}^{+}\,,\,\,D^{+}_{s}D_{s}^{*+}\,, (25)

the available phase spaces are large, the decays are kinematically facilitated, the doubly charmed tetraquark states with the JP=1superscript𝐽𝑃superscript1J^{P}=1^{-} should have large widths. We can search for the doubly charmed tetraquark states in those decay channels in the future.

T2(GeV2)superscript𝑇2superscriptGeV2T^{2}(\rm{GeV}^{2}) s0(GeV)subscript𝑠0GeV\sqrt{s_{0}}(\rm{GeV}) μ(GeV)𝜇GeV\mu(\rm{GeV}) pole M(GeV)𝑀GeVM(\rm{GeV}) λ(GeV5)𝜆superscriptGeV5\lambda(\rm{GeV}^{5})
ccu¯d¯(0+)𝑐𝑐¯𝑢¯𝑑superscript0cc\bar{u}\bar{d}(0^{+}) 2.42.82.42.82.4-2.8 4.40±0.10plus-or-minus4.400.104.40\pm 0.10 1.2 (3863)%percent3863(38-63)\% 3.87±0.09plus-or-minus3.870.093.87\pm 0.09 (3.90±0.63)×102plus-or-minus3.900.63superscript102(3.90\pm 0.63)\times 10^{-2}
ccu¯s¯(0+)𝑐𝑐¯𝑢¯𝑠superscript0cc\bar{u}\bar{s}(0^{+}) 2.63.02.63.02.6-3.0 4.50±0.10plus-or-minus4.500.104.50\pm 0.10 1.3 (3862)%percent3862(38-62)\% 3.94±0.10plus-or-minus3.940.103.94\pm 0.10 (4.92±0.89)×102plus-or-minus4.920.89superscript102(4.92\pm 0.89)\times 10^{-2}
ccs¯s¯(0+)𝑐𝑐¯𝑠¯𝑠superscript0cc\bar{s}\bar{s}(0^{+}) 2.63.02.63.02.6-3.0 4.55±0.10plus-or-minus4.550.104.55\pm 0.10 1.3 (3963)%percent3963(39-63)\% 3.99±0.10plus-or-minus3.990.103.99\pm 0.10 (5.31±0.99)×102plus-or-minus5.310.99superscript102(5.31\pm 0.99)\times 10^{-2}
ccu¯d¯(1+)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}(1^{+}) 2.63.02.63.02.6-3.0 4.45±0.10plus-or-minus4.450.104.45\pm 0.10 1.3 (3962)%percent3962(39-62)\% 3.90±0.09plus-or-minus3.900.093.90\pm 0.09 (3.44±0.54)×102plus-or-minus3.440.54superscript102(3.44\pm 0.54)\times 10^{-2}
ccu¯s¯(1+)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}(1^{+}) 2.63.02.63.02.6-3.0 4.50±0.10plus-or-minus4.500.104.50\pm 0.10 1.3 (4064)%percent4064(40-64)\% 3.96±0.08plus-or-minus3.960.083.96\pm 0.08 (3.78±0.59)×102plus-or-minus3.780.59superscript102(3.78\pm 0.59)\times 10^{-2}
ccs¯s¯(1+)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}(1^{+}) 2.73.12.73.12.7-3.1 4.55±0.10plus-or-minus4.550.104.55\pm 0.10 1.3 (3962)%percent3962(39-62)\% 4.02±0.09plus-or-minus4.020.094.02\pm 0.09 (4.11±0.68)×102plus-or-minus4.110.68superscript102(4.11\pm 0.68)\times 10^{-2}
ccu¯d¯(2+)𝑐𝑐¯𝑢¯𝑑superscript2cc\bar{u}\bar{d}(2^{+}) 2.73.12.73.12.7-3.1 4.50±0.10plus-or-minus4.500.104.50\pm 0.10 1.4 (3962)%percent3962(39-62)\% 3.95±0.09plus-or-minus3.950.093.95\pm 0.09 (5.67±0.90)×102plus-or-minus5.670.90superscript102(5.67\pm 0.90)\times 10^{-2}
ccu¯s¯(2+)𝑐𝑐¯𝑢¯𝑠superscript2cc\bar{u}\bar{s}(2^{+}) 2.83.22.83.22.8-3.2 4.55±0.10plus-or-minus4.550.104.55\pm 0.10 1.4 (3860)%percent3860(38-60)\% 4.01±0.09plus-or-minus4.010.094.01\pm 0.09 (6.27±1.02)×102plus-or-minus6.271.02superscript102(6.27\pm 1.02)\times 10^{-2}
ccs¯s¯(2+)𝑐𝑐¯𝑠¯𝑠superscript2cc\bar{s}\bar{s}(2^{+}) 2.83.22.83.22.8-3.2 4.60±0.10plus-or-minus4.600.104.60\pm 0.10 1.4 (3961)%percent3961(39-61)\% 4.06±0.09plus-or-minus4.060.094.06\pm 0.09 (6.78±1.12)×102plus-or-minus6.781.12superscript102(6.78\pm 1.12)\times 10^{-2}
ccu¯d¯(1)𝑐𝑐¯𝑢¯𝑑superscript1cc\bar{u}\bar{d}(1^{-}) 3.33.93.33.93.3-3.9 5.20±0.10plus-or-minus5.200.105.20\pm 0.10 2.9 (5073)%percent5073(50-73)\% 4.66±0.10plus-or-minus4.660.104.66\pm 0.10 (1.31±0.17)×101plus-or-minus1.310.17superscript101(1.31\pm 0.17)\times 10^{-1}
ccu¯s¯(1)𝑐𝑐¯𝑢¯𝑠superscript1cc\bar{u}\bar{s}(1^{-}) 3.44.03.44.03.4-4.0 5.25±0.10plus-or-minus5.250.105.25\pm 0.10 2.9 (4971)%percent4971(49-71)\% 4.73±0.11plus-or-minus4.730.114.73\pm 0.11 (1.40±0.19)×101plus-or-minus1.400.19superscript101(1.40\pm 0.19)\times 10^{-1}
ccs¯s¯(1)𝑐𝑐¯𝑠¯𝑠superscript1cc\bar{s}\bar{s}(1^{-}) 3.74.33.74.33.7-4.3 5.30±0.10plus-or-minus5.300.105.30\pm 0.10 2.9 (4972)%percent4972(49-72)\% 4.78±0.11plus-or-minus4.780.114.78\pm 0.11 (1.48±0.19)×101plus-or-minus1.480.19superscript101(1.48\pm 0.19)\times 10^{-1}
Table 1: The Borel parameters (Borel windows), continuum threshold parameters, optimal energy scales, pole contributions, masses and pole residues for the doubly charmed tetraquark states.
Refer to caption
Figure 6: The masses of the axialvector tetraquark states with variations of the Borel parameter T2superscript𝑇2T^{2} for the energy scale μ=1.3GeV𝜇1.3GeV\mu=1.3\,\rm{GeV} and continuum threshold parameter s0=4.45GeVsubscript𝑠04.45GeV\sqrt{s_{0}}=4.45\,\rm{GeV}, where the A𝐴A, B𝐵B and C𝐶C denote the CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type tetraquark state cc¯ud¯𝑐¯𝑐𝑢¯𝑑c\bar{c}u\bar{d}, Cγμγ5Ctensor-product𝐶subscript𝛾𝜇subscript𝛾5𝐶C\gamma_{\mu}\otimes\gamma_{5}C type tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d} and CγμγνCtensor-product𝐶subscript𝛾𝜇subscript𝛾𝜈𝐶C\gamma_{\mu}\otimes\gamma_{\nu}C type tetraquark state ccu¯d¯𝑐𝑐¯𝑢¯𝑑cc\bar{u}\bar{d}, respectively.

4 Conclusion

In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, vector, tensor doubly charmed tetraquark states, and study them with the QCD sum rules in a systematic way. In calculations, we carry out the operator product expansion up to the vacuum condensates of dimension 10 consistently, then obtain the QCD spectral densities through dispersion relation, and extract the masses and pole residues in the Borel windows at the optimal energy scales of the QCD spectral densities, which are determined by the energy scale formula with the refitted effective charm quark mass 𝕄csubscript𝕄𝑐{\mathbb{M}}_{c}. In the Borel windows, the pole dominance is satisfied and the operator product expansion is well convergent, so we expect to make reliable predictions. We can search for those doubly charmed tetraquark states in the Okubo-Zweig-Iizuka super-allowed strong decays to the charmed-meson pairs in the future.

Appendix

The explicit expressions of the QCD spectral densities ρu¯d¯;0(s)subscript𝜌¯𝑢¯𝑑0𝑠\rho_{\bar{u}\bar{d};0}(s), ρu¯s¯;0(s)subscript𝜌¯𝑢¯𝑠0𝑠\rho_{\bar{u}\bar{s};0}(s), ρs¯s¯;0(s)subscript𝜌¯𝑠¯𝑠0𝑠\rho_{\bar{s}\bar{s};0}(s), ρu¯d¯;1;A(s)subscript𝜌¯𝑢¯𝑑1𝐴𝑠\rho_{\bar{u}\bar{d};1;A}(s), ρu¯s¯;1;A(s)subscript𝜌¯𝑢¯𝑠1𝐴𝑠\rho_{\bar{u}\bar{s};1;A}(s), ρs¯s¯;1;A(s)subscript𝜌¯𝑠¯𝑠1𝐴𝑠\rho_{\bar{s}\bar{s};1;A}(s), ρu¯d¯;1;V(s)subscript𝜌¯𝑢¯𝑑1𝑉𝑠\rho_{\bar{u}\bar{d};1;V}(s), ρu¯s¯;1;V(s)subscript𝜌¯𝑢¯𝑠1𝑉𝑠\rho_{\bar{u}\bar{s};1;V}(s), ρs¯s¯;1;V(s)subscript𝜌¯𝑠¯𝑠1𝑉𝑠\rho_{\bar{s}\bar{s};1;V}(s), ρu¯d¯;2(s)subscript𝜌¯𝑢¯𝑑2𝑠\rho_{\bar{u}\bar{d};2}(s), ρu¯s¯;2(s)subscript𝜌¯𝑢¯𝑠2𝑠\rho_{\bar{u}\bar{s};2}(s), ρs¯s¯;2(s)subscript𝜌¯𝑠¯𝑠2𝑠\rho_{\bar{s}\bar{s};2}(s),

ρu¯s¯;0(s)subscript𝜌¯𝑢¯𝑠0𝑠\displaystyle\rho_{\bar{u}\bar{s};0}(s) =\displaystyle= ρ0;0(s)+ρ3;0(s)+ρ4;0(s)+ρ5;0(s)+ρ6;0(s)+ρ8;0(s)+ρ10;0(s),subscript𝜌00𝑠subscript𝜌30𝑠subscript𝜌40𝑠subscript𝜌50𝑠subscript𝜌60𝑠subscript𝜌80𝑠subscript𝜌100𝑠\displaystyle\rho_{0;0}(s)+\rho_{3;0}(s)+\rho_{4;0}(s)+\rho_{5;0}(s)+\rho_{6;0}(s)+\rho_{8;0}(s)+\rho_{10;0}(s)\,,
ρu¯s¯;1;A(s)subscript𝜌¯𝑢¯𝑠1𝐴𝑠\displaystyle\rho_{\bar{u}\bar{s};1;A}(s) =\displaystyle= ρ0;1;A(s)+ρ3;1;A(s)+ρ4;1;A(s)+ρ5;1;A(s)+ρ6;1;A(s)+ρ8;1;A(s)+ρ10;1;A(s),subscript𝜌01𝐴𝑠subscript𝜌31𝐴𝑠subscript𝜌41𝐴𝑠subscript𝜌51𝐴𝑠subscript𝜌61𝐴𝑠subscript𝜌81𝐴𝑠subscript𝜌101𝐴𝑠\displaystyle\rho_{0;1;A}(s)+\rho_{3;1;A}(s)+\rho_{4;1;A}(s)+\rho_{5;1;A}(s)+\rho_{6;1;A}(s)+\rho_{8;1;A}(s)+\rho_{10;1;A}(s)\,,
ρu¯s¯;1;V(s)subscript𝜌¯𝑢¯𝑠1𝑉𝑠\displaystyle\rho_{\bar{u}\bar{s};1;V}(s) =\displaystyle= ρ0;1;V(s)+ρ3;1;V(s)+ρ4;1;V(s)+ρ5;1;V(s)+ρ6;1;V(s)+ρ8;1;V(s)+ρ10;1;V(s),subscript𝜌01𝑉𝑠subscript𝜌31𝑉𝑠subscript𝜌41𝑉𝑠subscript𝜌51𝑉𝑠subscript𝜌61𝑉𝑠subscript𝜌81𝑉𝑠subscript𝜌101𝑉𝑠\displaystyle\rho_{0;1;V}(s)+\rho_{3;1;V}(s)+\rho_{4;1;V}(s)+\rho_{5;1;V}(s)+\rho_{6;1;V}(s)+\rho_{8;1;V}(s)+\rho_{10;1;V}(s)\,,
ρu¯s¯;2(s)subscript𝜌¯𝑢¯𝑠2𝑠\displaystyle\rho_{\bar{u}\bar{s};2}(s) =\displaystyle= ρ0;2(s)+ρ3;2(s)+ρ4;2(s)+ρ5;2(s)+ρ6;2(s)+ρ8;2(s)+ρ10;2(s),subscript𝜌02𝑠subscript𝜌32𝑠subscript𝜌42𝑠subscript𝜌52𝑠subscript𝜌62𝑠subscript𝜌82𝑠subscript𝜌102𝑠\displaystyle\rho_{0;2}(s)+\rho_{3;2}(s)+\rho_{4;2}(s)+\rho_{5;2}(s)+\rho_{6;2}(s)+\rho_{8;2}(s)+\rho_{10;2}(s)\,, (26)
ρu¯d¯;0(s)subscript𝜌¯𝑢¯𝑑0𝑠\displaystyle\rho_{\bar{u}\bar{d};0}(s) =\displaystyle= ρu¯s¯;0(s)ms0,s¯sq¯q,s¯gsσGsq¯gsσGq,evaluated-atsubscript𝜌¯𝑢¯𝑠0𝑠formulae-sequencesubscript𝑚𝑠0formulae-sequencedelimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞\displaystyle\rho_{\bar{u}\bar{s};0}(s)\mid_{m_{s}\to 0,\,\langle\bar{s}s\rangle\to\langle\bar{q}q\rangle,\,\langle\bar{s}g_{s}\sigma Gs\rangle\to\langle\bar{q}g_{s}\sigma Gq\rangle}\,,
ρu¯d¯;1;A(s)subscript𝜌¯𝑢¯𝑑1𝐴𝑠\displaystyle\rho_{\bar{u}\bar{d};1;A}(s) =\displaystyle= ρu¯s¯;1;A(s)ms0,s¯sq¯q,s¯gsσGsq¯gsσGq,evaluated-atsubscript𝜌¯𝑢¯𝑠1𝐴𝑠formulae-sequencesubscript𝑚𝑠0formulae-sequencedelimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞\displaystyle\rho_{\bar{u}\bar{s};1;A}(s)\mid_{m_{s}\to 0,\,\langle\bar{s}s\rangle\to\langle\bar{q}q\rangle,\,\langle\bar{s}g_{s}\sigma Gs\rangle\to\langle\bar{q}g_{s}\sigma Gq\rangle}\,,
ρu¯d¯;1;V(s)subscript𝜌¯𝑢¯𝑑1𝑉𝑠\displaystyle\rho_{\bar{u}\bar{d};1;V}(s) =\displaystyle= ρu¯s¯;1;V(s)ms0,s¯sq¯q,s¯gsσGsq¯gsσGq,evaluated-atsubscript𝜌¯𝑢¯𝑠1𝑉𝑠formulae-sequencesubscript𝑚𝑠0formulae-sequencedelimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞\displaystyle\rho_{\bar{u}\bar{s};1;V}(s)\mid_{m_{s}\to 0,\,\langle\bar{s}s\rangle\to\langle\bar{q}q\rangle,\,\langle\bar{s}g_{s}\sigma Gs\rangle\to\langle\bar{q}g_{s}\sigma Gq\rangle}\,,
ρu¯d¯;2(s)subscript𝜌¯𝑢¯𝑑2𝑠\displaystyle\rho_{\bar{u}\bar{d};2}(s) =\displaystyle= ρu¯s¯;2(s)ms0,s¯sq¯q,s¯gsσGsq¯gsσGq,evaluated-atsubscript𝜌¯𝑢¯𝑠2𝑠formulae-sequencesubscript𝑚𝑠0formulae-sequencedelimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞\displaystyle\rho_{\bar{u}\bar{s};2}(s)\mid_{m_{s}\to 0,\,\langle\bar{s}s\rangle\to\langle\bar{q}q\rangle,\,\langle\bar{s}g_{s}\sigma Gs\rangle\to\langle\bar{q}g_{s}\sigma Gq\rangle}\,, (27)
ρs¯s¯;0(s)subscript𝜌¯𝑠¯𝑠0𝑠\displaystyle\rho_{\bar{s}\bar{s};0}(s) =\displaystyle= ρu¯s¯;0(s)ms2ms,q¯qs¯s,q¯gsσGqs¯gsσGs,evaluated-atsubscript𝜌¯𝑢¯𝑠0𝑠formulae-sequencesubscript𝑚𝑠2subscript𝑚𝑠formulae-sequencedelimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠\displaystyle\rho_{\bar{u}\bar{s};0}(s)\mid_{m_{s}\to 2m_{s},\,\langle\bar{q}q\rangle\to\langle\bar{s}s\rangle,\,\langle\bar{q}g_{s}\sigma Gq\rangle\to\langle\bar{s}g_{s}\sigma Gs\rangle}\,,
ρs¯s¯;1;A(s)subscript𝜌¯𝑠¯𝑠1𝐴𝑠\displaystyle\rho_{\bar{s}\bar{s};1;A}(s) =\displaystyle= ρu¯s¯;1;A(s)ms2ms,q¯qs¯s,q¯gsσGqs¯gsσGs,evaluated-atsubscript𝜌¯𝑢¯𝑠1𝐴𝑠formulae-sequencesubscript𝑚𝑠2subscript𝑚𝑠formulae-sequencedelimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠\displaystyle\rho_{\bar{u}\bar{s};1;A}(s)\mid_{m_{s}\to 2m_{s},\,\langle\bar{q}q\rangle\to\langle\bar{s}s\rangle,\,\langle\bar{q}g_{s}\sigma Gq\rangle\to\langle\bar{s}g_{s}\sigma Gs\rangle}\,,
ρs¯s¯;1;V(s)subscript𝜌¯𝑠¯𝑠1𝑉𝑠\displaystyle\rho_{\bar{s}\bar{s};1;V}(s) =\displaystyle= ρu¯s¯;1;V(s)ms2ms,q¯qs¯s,q¯gsσGqs¯gsσGs,evaluated-atsubscript𝜌¯𝑢¯𝑠1𝑉𝑠formulae-sequencesubscript𝑚𝑠2subscript𝑚𝑠formulae-sequencedelimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠\displaystyle\rho_{\bar{u}\bar{s};1;V}(s)\mid_{m_{s}\to 2m_{s},\,\langle\bar{q}q\rangle\to\langle\bar{s}s\rangle,\,\langle\bar{q}g_{s}\sigma Gq\rangle\to\langle\bar{s}g_{s}\sigma Gs\rangle}\,,
ρs¯s¯;2(s)subscript𝜌¯𝑠¯𝑠2𝑠\displaystyle\rho_{\bar{s}\bar{s};2}(s) =\displaystyle= ρu¯s¯;2(s)ms2ms,q¯qs¯s,q¯gsσGqs¯gsσGs,evaluated-atsubscript𝜌¯𝑢¯𝑠2𝑠formulae-sequencesubscript𝑚𝑠2subscript𝑚𝑠formulae-sequencedelimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠\displaystyle\rho_{\bar{u}\bar{s};2}(s)\mid_{m_{s}\to 2m_{s},\,\langle\bar{q}q\rangle\to\langle\bar{s}s\rangle,\,\langle\bar{q}g_{s}\sigma Gq\rangle\to\langle\bar{s}g_{s}\sigma Gs\rangle}\,, (28)
ρ0;0(s)subscript𝜌00𝑠\displaystyle\rho_{0;0}(s) =\displaystyle= 164π6yiyf𝑑yzi1y𝑑zyz(1yz)2(sm¯c2)3(3sm¯c2)164superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐233𝑠superscriptsubscript¯𝑚𝑐2\displaystyle\frac{1}{64\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\left(3s-\overline{m}_{c}^{2}\right) (29)
+mc264π6yiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)3,superscriptsubscript𝑚𝑐264superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐23\displaystyle+\frac{m_{c}^{2}}{64\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\,,
ρ3;0(s)subscript𝜌30𝑠\displaystyle\rho_{3;0}(s) =\displaystyle= ms[q¯qs¯s]4π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(2sm¯c2)subscript𝑚𝑠delimited-[]delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠4superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\left[\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{4\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right) (30)
msmc2[4q¯qs¯s]8π4yiyf𝑑yzi1y𝑑z(sm¯c2),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]4delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠8superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\left[4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{8\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(s-\overline{m}_{c}^{2}\right)\,,
ρ4;0(s)subscript𝜌40𝑠\displaystyle\rho_{4;0}(s) =\displaystyle= mc296π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)2(3s2m¯c2)superscriptsubscript𝑚𝑐296superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧23𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{96\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{2}\left(3s-2\overline{m}_{c}^{2}\right) (31)
mc4192π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)2superscriptsubscript𝑚𝑐4192superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧2\displaystyle-\frac{m_{c}^{4}}{192\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{2}
+mc264π4αsGGπyiyf𝑑yzi1y𝑑z[(1y2+1z2)(1yz)21](sm¯c2),superscriptsubscript𝑚𝑐264superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]1superscript𝑦21superscript𝑧2superscript1𝑦𝑧21𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{64\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\left(\frac{1}{y^{2}}+\frac{1}{z^{2}}\right)(1-y-z)^{2}-1\right]\left(s-\overline{m}_{c}^{2}\right)\,,
ρ5;0(s)subscript𝜌50𝑠\displaystyle\rho_{5;0}(s) =\displaystyle= ms[3q¯gsσGq2s¯gsσGs]48π4yiyf𝑑yy(1y)(3s2m~c2)subscript𝑚𝑠delimited-[]3delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠48superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠2superscriptsubscript~𝑚𝑐2\displaystyle\frac{m_{s}\left[3\langle\bar{q}g_{s}\sigma Gq\rangle-2\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{48\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-2\widetilde{m}_{c}^{2}\right) (32)
+msmc2[6q¯gsσGqs¯gsσGs]48π4yiyf𝑑ysubscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]6delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠48superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦\displaystyle+\frac{m_{s}m_{c}^{2}\left[6\langle\bar{q}g_{s}\sigma Gq\rangle-\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{48\pi^{4}}\int_{y_{i}}^{y_{f}}dy
msq¯gsσGq64π4yiyf𝑑yzi1y𝑑z(y+z)(3s2m¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞64superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧3𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{64\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(y+z)\left(3s-2\overline{m}_{c}^{2}\right)
msmc2q¯gsσGq32π4yiyf𝑑yzi1y𝑑z(1y+1z),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞32superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦1𝑧\displaystyle-\frac{m_{s}m_{c}^{2}\langle\bar{q}g_{s}\sigma Gq\rangle}{32\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y}+\frac{1}{z}\right)\,,
ρ6;0(s)subscript𝜌60𝑠\displaystyle\rho_{6;0}(s) =\displaystyle= q¯qs¯sπ2yiyf𝑑yy(1y)s,delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠\displaystyle\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)s\,, (33)
ρ8;0(s)subscript𝜌80𝑠\displaystyle\rho_{8;0}(s) =\displaystyle= s¯sq¯gsσGq+q¯qs¯gsσGs4π2yiyf𝑑yy(1y)[2+(2s+s2T2)δ(sm~c2)]delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠4superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦delimited-[]22𝑠superscript𝑠2superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{4\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left[2+\left(2s+\frac{s^{2}}{T^{2}}\right)\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right] (34)
+s¯sq¯gsσGq+q¯qs¯gsσGs48π2yiyf𝑑y[2+3sδ(sm~c2)],delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠48superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦delimited-[]23𝑠𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{48\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\left[2+3s\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right]\,,
ρ10;0(s)subscript𝜌100𝑠\displaystyle\rho_{10;0}(s) =\displaystyle= q¯gsσGqs¯gsσGs16π2yiyf𝑑yy(1y)(2+2sT2+s2T4+s3T6)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠16superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦22𝑠superscript𝑇2superscript𝑠2superscript𝑇4superscript𝑠3superscript𝑇6𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{16\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\,\left(2+\frac{2s}{T^{2}}+\frac{s^{2}}{T^{4}}+\frac{s^{3}}{T^{6}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right) (35)
q¯gsσGqs¯gsσGs96π2yiyf𝑑y(2+2sT2+3s2T4)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠96superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦22𝑠superscript𝑇23superscript𝑠2superscript𝑇4𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{96\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(2+\frac{2s}{T^{2}}+\frac{3s^{2}}{T^{4}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)
+11q¯gsσGqs¯gsσGs768π2yiyf𝑑y(1+5sT2)δ(sm~c2),11delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠768superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦15𝑠superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{11\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{768\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(1+\frac{5s}{T^{2}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,
ρ0;1;A(s)subscript𝜌01𝐴𝑠\displaystyle\rho_{0;1;A}(s) =\displaystyle= 1384π6yiyf𝑑yzi1y𝑑zyz(1yz)3(sm¯c2)2(21s214sm¯c2+m¯c4)1384superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐2221superscript𝑠214𝑠superscriptsubscript¯𝑚𝑐2superscriptsubscript¯𝑚𝑐4\displaystyle\frac{1}{384\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(21s^{2}-14s\overline{m}_{c}^{2}+\overline{m}_{c}^{4}\right) (36)
1384π6yiyf𝑑yzi1y𝑑zyz(1yz)2(sm¯c2)3(3sm¯c2)1384superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐233𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{384\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\left(3s-\overline{m}_{c}^{2}\right)
+mc2576π6yiyf𝑑yzi1y𝑑z(1yz)3(sm¯c2)2(7sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐227𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(7s-\overline{m}_{c}^{2}\right)
+mc2192π6yiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)3,superscriptsubscript𝑚𝑐2192superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐23\displaystyle+\frac{m_{c}^{2}}{192\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\,,
ρ3;1;A(s)subscript𝜌31𝐴𝑠\displaystyle\rho_{3;1;A}(s) =\displaystyle= mss¯s24π4yiyf𝑑yzi1y𝑑zyz(1yz)(25s224sm¯c2+3m¯c4)subscript𝑚𝑠delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧25superscript𝑠224𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle\frac{m_{s}\langle\bar{s}s\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(25s^{2}-24s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right) (37)
msq¯q24π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(5sm¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞𝑞24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐25𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{q}q\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(5s-\overline{m}_{c}^{2}\right)
ms[2q¯q+s¯s]24π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(2sm¯c2)subscript𝑚𝑠delimited-[]2delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\left[2\langle\bar{q}q\rangle+\langle\bar{s}s\rangle\right]}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right)
msmc2[6q¯qs¯s]24π4yiyf𝑑yzi1y𝑑z(sm¯c2)subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]6delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\left[6\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(s-\overline{m}_{c}^{2}\right)
+msmc2s¯s24π4yiyf𝑑yzi1y𝑑z(1yz)(3sm¯c2),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧3𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}m_{c}^{2}\langle\bar{s}s\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left(3s-\overline{m}_{c}^{2}\right)\,,
ρ4;1;A(s)subscript𝜌41𝐴𝑠\displaystyle\rho_{4;1;A}(s) =\displaystyle= mc2288π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)3[4sm¯c2+2s23δ(sm¯c2)]superscriptsubscript𝑚𝑐2288superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧3delimited-[]4𝑠superscriptsubscript¯𝑚𝑐22superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{288\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{3}\left[4s-\overline{m}_{c}^{2}+\frac{2s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (38)
+mc2576π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)2(3s2m¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧23𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{2}\left(3s-2\overline{m}_{c}^{2}\right)
mc4864π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)3[12+sδ(sm¯c2)]superscriptsubscript𝑚𝑐4864superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧3delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{4}}{864\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{3}\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
+mc2576π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)[1+(1y2+1z2)(1yz)2](3sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧delimited-[]11superscript𝑦21superscript𝑧2superscript1𝑦𝑧23𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left[1+\left(\frac{1}{y^{2}}+\frac{1}{z^{2}}\right)(1-y-z)^{2}\right]\left(3s-\overline{m}_{c}^{2}\right)
mc4576π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)2superscriptsubscript𝑚𝑐4576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧2\displaystyle-\frac{m_{c}^{4}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{2}
+mc2576π4αsGGπyiyf𝑑yzi1y𝑑z[(3y2+3z2)(1yz)25](sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]3superscript𝑦23superscript𝑧2superscript1𝑦𝑧25𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\left(\frac{3}{y^{2}}+\frac{3}{z^{2}}\right)(1-y-z)^{2}-5\right]\left(s-\overline{m}_{c}^{2}\right)
1384π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)(2sm¯c2)1384superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{384\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right)
+1576π4αsGGπyiyf𝑑yzi1y𝑑zyz(1yz)(25s224sm¯c2+3m¯c4)1576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧25superscript𝑠224𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle+\frac{1}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(25s^{2}-24s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right)
1576π4αsGGπyiyf𝑑yzi1y𝑑zyz(sm¯c2)(11s4m¯c2),1576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐211𝑠4superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(11s-4\overline{m}_{c}^{2}\right)\,,
ρ5;1;A(s)subscript𝜌51𝐴𝑠\displaystyle\rho_{5;1;A}(s) =\displaystyle= mss¯gsσGs24π4yiyf𝑑yzi1y𝑑zyz[4sm¯c2+2s23δ(sm¯c2)]subscript𝑚𝑠delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧delimited-[]4𝑠superscriptsubscript¯𝑚𝑐22superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{s}g_{s}\sigma Gs\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left[4s-\overline{m}_{c}^{2}+\frac{2s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (39)
+msq¯gsσGq48π4yiyf𝑑yy(1y)(3sm~c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞48superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{48\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-\widetilde{m}_{c}^{2}\right)
+ms[3q¯gsσGq+s¯gsσGs]144π4yiyf𝑑yy(1y)(3s2m~c2)subscript𝑚𝑠delimited-[]3delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠2superscriptsubscript~𝑚𝑐2\displaystyle+\frac{m_{s}\left[3\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{144\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-2\widetilde{m}_{c}^{2}\right)
msmc2s¯gsσGs72π4yiyf𝑑yzi1y𝑑z[12+sδ(sm¯c2)]subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠72superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\langle\bar{s}g_{s}\sigma Gs\rangle}{72\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
+msmc2[9q¯gsσGqs¯gsσGs]144π4yiyf𝑑ysubscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]9delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦\displaystyle+\frac{m_{s}m_{c}^{2}\left[9\langle\bar{q}g_{s}\sigma Gq\rangle-\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{144\pi^{4}}\int_{y_{i}}^{y_{f}}dy
msq¯gsσGq192π4yiyf𝑑yzi1y𝑑z(y+z)(2sm¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞192superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧2𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{192\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(y+z)\left(2s-\overline{m}_{c}^{2}\right)
msmc2q¯gsσGq192π4yiyf𝑑yzi1y𝑑z(1y+1z),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞192superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦1𝑧\displaystyle-\frac{m_{s}m_{c}^{2}\langle\bar{q}g_{s}\sigma Gq\rangle}{192\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y}+\frac{1}{z}\right)\,,
ρ6;1;A(s)subscript𝜌61𝐴𝑠\displaystyle\rho_{6;1;A}(s) =\displaystyle= 2q¯qs¯s3π2yiyf𝑑yy(1y)s,2delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠3superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠\displaystyle\frac{2\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)s\,, (40)
ρ8;1;A(s)subscript𝜌81𝐴𝑠\displaystyle\rho_{8;1;A}(s) =\displaystyle= s¯sq¯gsσGq+q¯qs¯gsσGs12π2yiyf𝑑yy(1y)[3+(4s+2s2T2)δ(sm~c2)]delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠12superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦delimited-[]34𝑠2superscript𝑠2superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{12\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left[3+\left(4s+\frac{2s^{2}}{T^{2}}\right)\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right] (41)
+s¯sq¯gsσGq+q¯qs¯gsσGs144π2yiyf𝑑y[1+2sδ(sm~c2)],delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦delimited-[]12𝑠𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{144\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\left[1+2s\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right]\,,
ρ10;1;A(s)subscript𝜌101𝐴𝑠\displaystyle\rho_{10;1;A}(s) =\displaystyle= q¯gsσGqs¯gsσGs24π2yiyf𝑑yy(1y)(sT2+s2T4+s3T6)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠24superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠superscript𝑇2superscript𝑠2superscript𝑇4superscript𝑠3superscript𝑇6𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{24\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\,\left(\frac{s}{T^{2}}+\frac{s^{2}}{T^{4}}+\frac{s^{3}}{T^{6}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right) (42)
q¯gsσGqs¯gsσGs288π2yiyf𝑑y(sT2+2s2T4)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠288superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑠superscript𝑇22superscript𝑠2superscript𝑇4𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{288\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(\frac{s}{T^{2}}+\frac{2s^{2}}{T^{4}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)
11q¯gsσGqs¯gsσGs768π2yiyf𝑑ysT2δ(sm~c2),11delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠768superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑠superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{11\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{768\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\frac{s}{T^{2}}\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,
ρ0;1;V(s)subscript𝜌01𝑉𝑠\displaystyle\rho_{0;1;V}(s) =\displaystyle= 1384π6yiyf𝑑yzi1y𝑑zyz(1yz)3(sm¯c2)2(21s214sm¯c2+m¯c4)1384superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐2221superscript𝑠214𝑠superscriptsubscript¯𝑚𝑐2superscriptsubscript¯𝑚𝑐4\displaystyle\frac{1}{384\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(21s^{2}-14s\overline{m}_{c}^{2}+\overline{m}_{c}^{4}\right) (43)
1384π6yiyf𝑑yzi1y𝑑zyz(1yz)2(sm¯c2)3(3sm¯c2)1384superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐233𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{384\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\left(3s-\overline{m}_{c}^{2}\right)
+mc2576π6yiyf𝑑yzi1y𝑑z(1yz)3(sm¯c2)2(7sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐227𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(7s-\overline{m}_{c}^{2}\right)
mc296π6yiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)3,superscriptsubscript𝑚𝑐296superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐23\displaystyle-\frac{m_{c}^{2}}{96\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\,,
ρ3;1;V(s)subscript𝜌31𝑉𝑠\displaystyle\rho_{3;1;V}(s) =\displaystyle= mss¯s24π4yiyf𝑑yzi1y𝑑zyz(1yz)(25s224sm¯c2+3m¯c4)subscript𝑚𝑠delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧25superscript𝑠224𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle\frac{m_{s}\langle\bar{s}s\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(25s^{2}-24s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right) (44)
msq¯q24π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(5sm¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞𝑞24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐25𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{q}q\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(5s-\overline{m}_{c}^{2}\right)
+ms[4q¯qs¯s]24π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(2sm¯c2)subscript𝑚𝑠delimited-[]4delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}\left[4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right)
+msmc2[4q¯qs¯s]12π4yiyf𝑑yzi1y𝑑z(sm¯c2)subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]4delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠12superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}m_{c}^{2}\left[4\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{12\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(s-\overline{m}_{c}^{2}\right)
+msmc2s¯s24π4yiyf𝑑yzi1y𝑑z(1yz)(3sm¯c2),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧3𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}m_{c}^{2}\langle\bar{s}s\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left(3s-\overline{m}_{c}^{2}\right)\,,
ρ4;1;V(s)subscript𝜌41𝑉𝑠\displaystyle\rho_{4;1;V}(s) =\displaystyle= mc2288π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)3[4sm¯c2+2s23δ(sm¯c2)]superscriptsubscript𝑚𝑐2288superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧3delimited-[]4𝑠superscriptsubscript¯𝑚𝑐22superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{288\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{3}\left[4s-\overline{m}_{c}^{2}+\frac{2s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (45)
+mc2576π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)2(3s2m¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧23𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{2}\left(3s-2\overline{m}_{c}^{2}\right)
mc4864π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)3[12+sδ(sm¯c2)]superscriptsubscript𝑚𝑐4864superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧3delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{4}}{864\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{3}\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
+mc2576π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)[1+(1y2+1z2)(1yz)2](3sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧delimited-[]11superscript𝑦21superscript𝑧2superscript1𝑦𝑧23𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left[1+\left(\frac{1}{y^{2}}+\frac{1}{z^{2}}\right)(1-y-z)^{2}\right]\left(3s-\overline{m}_{c}^{2}\right)
+mc4576π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)2superscriptsubscript𝑚𝑐4576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧2\displaystyle+\frac{m_{c}^{4}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{2}
mc2576π4αsGGπyiyf𝑑yzi1y𝑑z[(3y2+3z2)(1yz)24](sm¯c2)superscriptsubscript𝑚𝑐2576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]3superscript𝑦23superscript𝑧2superscript1𝑦𝑧24𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\left(\frac{3}{y^{2}}+\frac{3}{z^{2}}\right)(1-y-z)^{2}-4\right]\left(s-\overline{m}_{c}^{2}\right)
+1384π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)(2sm¯c2)1384superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{1}{384\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right)
+1576π4αsGGπyiyf𝑑yzi1y𝑑zyz(1yz)(25s224sm¯c2+3m¯c4)1576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧25superscript𝑠224𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle+\frac{1}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(25s^{2}-24s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right)
+1576π4αsGGπyiyf𝑑yzi1y𝑑zyz(sm¯c2)(s2m¯c2),1576superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐2𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{1}{576\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(s-2\overline{m}_{c}^{2}\right)\,,
ρ5;1;V(s)subscript𝜌51𝑉𝑠\displaystyle\rho_{5;1;V}(s) =\displaystyle= mss¯gsσGs24π4yiyf𝑑yzi1y𝑑zyz[4sm¯c2+2s23δ(sm¯c2)]subscript𝑚𝑠delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠24superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧delimited-[]4𝑠superscriptsubscript¯𝑚𝑐22superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{s}g_{s}\sigma Gs\rangle}{24\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left[4s-\overline{m}_{c}^{2}+\frac{2s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (46)
+msq¯gsσGq48π4yiyf𝑑yy(1y)(3sm~c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞48superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{48\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-\widetilde{m}_{c}^{2}\right)
ms[6q¯gsσGqs¯gsσGs]144π4yiyf𝑑yy(1y)(3s2m~c2)subscript𝑚𝑠delimited-[]6delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠2superscriptsubscript~𝑚𝑐2\displaystyle-\frac{m_{s}\left[6\langle\bar{q}g_{s}\sigma Gq\rangle-\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{144\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-2\widetilde{m}_{c}^{2}\right)
msmc2s¯gsσGs72π4yiyf𝑑yzi1y𝑑z[12+sδ(sm¯c2)]subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠72superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\langle\bar{s}g_{s}\sigma Gs\rangle}{72\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
msmc2[9q¯gsσGq2s¯gsσGs]144π4yiyf𝑑ysubscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]9delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦\displaystyle-\frac{m_{s}m_{c}^{2}\left[9\langle\bar{q}g_{s}\sigma Gq\rangle-2\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{144\pi^{4}}\int_{y_{i}}^{y_{f}}dy
+msq¯gsσGq192π4yiyf𝑑yzi1y𝑑z(y+z)(sm¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞192superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{192\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(y+z)\left(s-\overline{m}_{c}^{2}\right)
+msmc2q¯gsσGq192π4yiyf𝑑yzi1y𝑑z(1y+1z),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞192superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦1𝑧\displaystyle+\frac{m_{s}m_{c}^{2}\langle\bar{q}g_{s}\sigma Gq\rangle}{192\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y}+\frac{1}{z}\right)\,,
ρ6;1;V(s)subscript𝜌61𝑉𝑠\displaystyle\rho_{6;1;V}(s) =\displaystyle= q¯qs¯s3π2yiyf𝑑yy(1y)s,delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠3superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠\displaystyle-\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)s\,, (47)
ρ8;1;V(s)subscript𝜌81𝑉𝑠\displaystyle\rho_{8;1;V}(s) =\displaystyle= s¯sq¯gsσGq+q¯qs¯gsσGs12π2yiyf𝑑yy(1y)[3+(2s+s2T2)δ(sm~c2)]delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠12superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦delimited-[]32𝑠superscript𝑠2superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{12\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left[3+\left(2s+\frac{s^{2}}{T^{2}}\right)\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right] (48)
s¯sq¯gsσGq+q¯qs¯gsσGs144π2yiyf𝑑y[1+sδ(sm~c2)],delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦delimited-[]1𝑠𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{144\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\left[1+s\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right]\,,
ρ10;1;V(s)subscript𝜌101𝑉𝑠\displaystyle\rho_{10;1;V}(s) =\displaystyle= q¯gsσGqs¯gsσGs48π2yiyf𝑑yy(1y)(6+4sT2+s2T4+s3T6)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠48superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦64𝑠superscript𝑇2superscript𝑠2superscript𝑇4superscript𝑠3superscript𝑇6𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{48\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\,\left(6+\frac{4s}{T^{2}}+\frac{s^{2}}{T^{4}}+\frac{s^{3}}{T^{6}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right) (49)
+q¯gsσGqs¯gsσGs288π2yiyf𝑑y(2+sT2+s2T4)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠288superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦2𝑠superscript𝑇2superscript𝑠2superscript𝑇4𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{288\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(2+\frac{s}{T^{2}}+\frac{s^{2}}{T^{4}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)
+11q¯gsσGqs¯gsσGs2304π2yiyf𝑑y(1+2sT2)δ(sm~c2),11delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠2304superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦12𝑠superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{11\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{2304\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(1+\frac{2s}{T^{2}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,
ρ0;2(s)subscript𝜌02𝑠\displaystyle\rho_{0;2}(s) =\displaystyle= 1960π6yiyf𝑑yzi1y𝑑zyz(1yz)3(sm¯c2)2(33s218sm¯c2+m¯c4)1960superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐2233superscript𝑠218𝑠superscriptsubscript¯𝑚𝑐2superscriptsubscript¯𝑚𝑐4\displaystyle\frac{1}{960\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(33s^{2}-18s\overline{m}_{c}^{2}+\overline{m}_{c}^{4}\right) (50)
+1480π6yiyf𝑑yzi1y𝑑zyz(1yz)2(sm¯c2)3(9s2m¯c2)1480superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐239𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{1}{480\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\left(9s-2\overline{m}_{c}^{2}\right)
+mc2288π6yiyf𝑑yzi1y𝑑z(1yz)3(sm¯c2)2(7sm¯c2)superscriptsubscript𝑚𝑐2288superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧3superscript𝑠superscriptsubscript¯𝑚𝑐227𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{288\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{3}\left(s-\overline{m}_{c}^{2}\right)^{2}\left(7s-\overline{m}_{c}^{2}\right)
+mc296π6yiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)3,superscriptsubscript𝑚𝑐296superscript𝜋6superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2superscript𝑠superscriptsubscript¯𝑚𝑐23\displaystyle+\frac{m_{c}^{2}}{96\pi^{6}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)^{3}\,,
ρ3;2(s)subscript𝜌32𝑠\displaystyle\rho_{3;2}(s) =\displaystyle= mss¯s60π4yiyf𝑑yzi1y𝑑zyz(1yz)(35s230sm¯c2+3m¯c4)subscript𝑚𝑠delimited-⟨⟩¯𝑠𝑠60superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧35superscript𝑠230𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle\frac{m_{s}\langle\bar{s}s\rangle}{60\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(35s^{2}-30s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right) (51)
ms[5q¯qs¯s]60π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(5sm¯c2)subscript𝑚𝑠delimited-[]5delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠60superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐25𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\left[5\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{60\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(5s-\overline{m}_{c}^{2}\right)
ms[10q¯q3s¯s]60π4yiyf𝑑yzi1y𝑑zyz(sm¯c2)(2sm¯c2)subscript𝑚𝑠delimited-[]10delimited-⟨⟩¯𝑞𝑞3delimited-⟨⟩¯𝑠𝑠60superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐22𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\left[10\langle\bar{q}q\rangle-3\langle\bar{s}s\rangle\right]}{60\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(2s-\overline{m}_{c}^{2}\right)
msmc2[6q¯qs¯s]12π4yiyf𝑑yzi1y𝑑z(sm¯c2)subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]6delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠12superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\left[6\langle\bar{q}q\rangle-\langle\bar{s}s\rangle\right]}{12\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(s-\overline{m}_{c}^{2}\right)
+msmc2s¯s12π4yiyf𝑑yzi1y𝑑z(1yz)(3sm¯c2),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠𝑠12superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧3𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}m_{c}^{2}\langle\bar{s}s\rangle}{12\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left(3s-\overline{m}_{c}^{2}\right)\,,
ρ4;2(s)subscript𝜌42𝑠\displaystyle\rho_{4;2}(s) =\displaystyle= mc2720π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)3[5sm¯c2+4s23δ(sm¯c2)]superscriptsubscript𝑚𝑐2720superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧3delimited-[]5𝑠superscriptsubscript¯𝑚𝑐24superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{720\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{3}\left[5s-\overline{m}_{c}^{2}+\frac{4s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (52)
mc21440π4αsGGπyiyf𝑑yzi1y𝑑z(zy2+yz2)(1yz)2(15s8m¯c2)superscriptsubscript𝑚𝑐21440superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑧superscript𝑦2𝑦superscript𝑧2superscript1𝑦𝑧215𝑠8superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{2}}{1440\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{z}{y^{2}}+\frac{y}{z^{2}}\right)(1-y-z)^{2}\left(15s-8\overline{m}_{c}^{2}\right)
mc4432π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)3[12+sδ(sm¯c2)]superscriptsubscript𝑚𝑐4432superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧3delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{c}^{4}}{432\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{3}\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
+mc2288π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)[1+(1y2+1z2)(1yz)2](3sm¯c2)superscriptsubscript𝑚𝑐2288superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦𝑧delimited-[]11superscript𝑦21superscript𝑧2superscript1𝑦𝑧23𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{288\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)\left[1+\left(\frac{1}{y^{2}}+\frac{1}{z^{2}}\right)(1-y-z)^{2}\right]\left(3s-\overline{m}_{c}^{2}\right)
mc4288π4αsGGπyiyf𝑑yzi1y𝑑z(1y3+1z3)(1yz)2superscriptsubscript𝑚𝑐4288superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1superscript𝑦31superscript𝑧3superscript1𝑦𝑧2\displaystyle-\frac{m_{c}^{4}}{288\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y^{3}}+\frac{1}{z^{3}}\right)(1-y-z)^{2}
+mc2288π4αsGGπyiyf𝑑yzi1y𝑑z[(3y2+3z2)(1yz)25](sm¯c2)superscriptsubscript𝑚𝑐2288superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]3superscript𝑦23superscript𝑧2superscript1𝑦𝑧25𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{c}^{2}}{288\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\left(\frac{3}{y^{2}}+\frac{3}{z^{2}}\right)(1-y-z)^{2}-5\right]\left(s-\overline{m}_{c}^{2}\right)
11440π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)3(15s215sm¯c2+2m¯c4)11440superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧315superscript𝑠215𝑠superscriptsubscript¯𝑚𝑐22superscriptsubscript¯𝑚𝑐4\displaystyle-\frac{1}{1440\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{3}\left(15s^{2}-15s\overline{m}_{c}^{2}+2\overline{m}_{c}^{4}\right)
1960π4αsGGπyiyf𝑑yzi1y𝑑z(1yz)2(sm¯c2)(3s2m¯c2)1960superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscript1𝑦𝑧2𝑠superscriptsubscript¯𝑚𝑐23𝑠2superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{960\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(1-y-z)^{2}\left(s-\overline{m}_{c}^{2}\right)\left(3s-2\overline{m}_{c}^{2}\right)
+11440π4αsGGπyiyf𝑑yzi1y𝑑zyz(1yz)(35s230sm¯c2+3m¯c4)11440superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧1𝑦𝑧35superscript𝑠230𝑠superscriptsubscript¯𝑚𝑐23superscriptsubscript¯𝑚𝑐4\displaystyle+\frac{1}{1440\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz(1-y-z)\left(35s^{2}-30s\overline{m}_{c}^{2}+3\overline{m}_{c}^{4}\right)
11440π4αsGGπyiyf𝑑yzi1y𝑑zyz(sm¯c2)(34s11m¯c2),11440superscript𝜋4delimited-⟨⟩subscript𝛼𝑠𝐺𝐺𝜋superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧𝑠superscriptsubscript¯𝑚𝑐234𝑠11superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{1}{1440\pi^{4}}\langle\frac{\alpha_{s}GG}{\pi}\rangle\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left(s-\overline{m}_{c}^{2}\right)\left(34s-11\overline{m}_{c}^{2}\right)\,,
ρ5;2(s)subscript𝜌52𝑠\displaystyle\rho_{5;2}(s) =\displaystyle= mss¯gsσGs60π4yiyf𝑑yzi1y𝑑zyz[5sm¯c2+4s23δ(sm¯c2)]subscript𝑚𝑠delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠60superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧delimited-[]5𝑠superscriptsubscript¯𝑚𝑐24superscript𝑠23𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}\langle\bar{s}g_{s}\sigma Gs\rangle}{60\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,yz\left[5s-\overline{m}_{c}^{2}+\frac{4s^{2}}{3}\delta\left(s-\overline{m}_{c}^{2}\right)\right] (53)
+ms[15q¯gsσGq2s¯gsσGs]360π4yiyf𝑑yy(1y)(3sm~c2)subscript𝑚𝑠delimited-[]15delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠360superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{m_{s}\left[15\langle\bar{q}g_{s}\sigma Gq\rangle-2\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{360\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-\widetilde{m}_{c}^{2}\right)
+ms[5q¯gsσGqs¯gsσGs]120π4yiyf𝑑yy(1y)(3s2m~c2)subscript𝑚𝑠delimited-[]5delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠120superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦3𝑠2superscriptsubscript~𝑚𝑐2\displaystyle+\frac{m_{s}\left[5\langle\bar{q}g_{s}\sigma Gq\rangle-\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{120\pi^{4}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left(3s-2\widetilde{m}_{c}^{2}\right)
+msmc2[9q¯gsσGqs¯gsσGs]72π4yiyf𝑑ysubscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-[]9delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠72superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦\displaystyle+\frac{m_{s}m_{c}^{2}\left[9\langle\bar{q}g_{s}\sigma Gq\rangle-\langle\bar{s}g_{s}\sigma Gs\rangle\right]}{72\pi^{4}}\int_{y_{i}}^{y_{f}}dy
msmc2s¯gsσGs36π4yiyf𝑑yzi1y𝑑z[12+sδ(sm¯c2)]subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠36superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧delimited-[]12𝑠𝛿𝑠superscriptsubscript¯𝑚𝑐2\displaystyle-\frac{m_{s}m_{c}^{2}\langle\bar{s}g_{s}\sigma Gs\rangle}{36\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left[\frac{1}{2}+s\,\delta\left(s-\overline{m}_{c}^{2}\right)\right]
+msq¯gsσGq96π4yiyf𝑑yzi1y𝑑z(y+z)(2sm¯c2)subscript𝑚𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞96superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧𝑦𝑧2𝑠superscriptsubscript¯𝑚𝑐2\displaystyle+\frac{m_{s}\langle\bar{q}g_{s}\sigma Gq\rangle}{96\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,(y+z)\left(2s-\overline{m}_{c}^{2}\right)
+msmc2q¯gsσGq96π4yiyf𝑑yzi1y𝑑z(1y+1z),subscript𝑚𝑠superscriptsubscript𝑚𝑐2delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞96superscript𝜋4superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧1𝑦1𝑧\displaystyle+\frac{m_{s}m_{c}^{2}\langle\bar{q}g_{s}\sigma Gq\rangle}{96\pi^{4}}\int_{y_{i}}^{y_{f}}dy\int_{z_{i}}^{1-y}dz\,\left(\frac{1}{y}+\frac{1}{z}\right)\,,
ρ6;2(s)subscript𝜌62𝑠\displaystyle\rho_{6;2}(s) =\displaystyle= 4q¯qs¯s3π2yiyf𝑑yy(1y)s,4delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠𝑠3superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠\displaystyle\frac{4\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)s\,, (54)
ρ8;2(s)subscript𝜌82𝑠\displaystyle\rho_{8;2}(s) =\displaystyle= s¯sq¯gsσGq+q¯qs¯gsσGs6π2yiyf𝑑yy(1y)[3+(4s+2s2T2)δ(sm~c2)]delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠6superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦delimited-[]34𝑠2superscript𝑠2superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{6\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\left[3+\left(4s+\frac{2s^{2}}{T^{2}}\right)\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right] (55)
s¯sq¯gsσGq+q¯qs¯gsσGs72π2yiyf𝑑y[1+2sδ(sm~c2)],delimited-⟨⟩¯𝑠𝑠delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑞𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠72superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦delimited-[]12𝑠𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle-\frac{\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle+\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{72\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\left[1+2s\,\delta\left(s-\widetilde{m}_{c}^{2}\right)\right]\,,
ρ10;2(s)subscript𝜌102𝑠\displaystyle\rho_{10;2}(s) =\displaystyle= q¯gsσGqs¯gsσGs12π2yiyf𝑑yy(1y)(sT2+s2T4+s3T6)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠12superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑦1𝑦𝑠superscript𝑇2superscript𝑠2superscript𝑇4superscript𝑠3superscript𝑇6𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{12\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,y(1-y)\,\left(\frac{s}{T^{2}}+\frac{s^{2}}{T^{4}}+\frac{s^{3}}{T^{6}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right) (56)
+q¯gsσGqs¯gsσGs144π2yiyf𝑑y(sT2+2s2T4)δ(sm~c2)delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠144superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑠superscript𝑇22superscript𝑠2superscript𝑇4𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{144\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\left(\frac{s}{T^{2}}+\frac{2s^{2}}{T^{4}}\right)\delta\left(s-\widetilde{m}_{c}^{2}\right)
+11q¯gsσGqs¯gsσGs1152π2yiyf𝑑ysT2δ(sm~c2),11delimited-⟨⟩¯𝑞subscript𝑔𝑠𝜎𝐺𝑞delimited-⟨⟩¯𝑠subscript𝑔𝑠𝜎𝐺𝑠1152superscript𝜋2superscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦𝑠superscript𝑇2𝛿𝑠superscriptsubscript~𝑚𝑐2\displaystyle+\frac{11\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle}{1152\pi^{2}}\int_{y_{i}}^{y_{f}}dy\,\,\frac{s}{T^{2}}\delta\left(s-\widetilde{m}_{c}^{2}\right)\,,

yf=1+14mc2/s2subscript𝑦𝑓114superscriptsubscript𝑚𝑐2𝑠2y_{f}=\frac{1+\sqrt{1-4m_{c}^{2}/s}}{2}, yi=114mc2/s2subscript𝑦𝑖114superscriptsubscript𝑚𝑐2𝑠2y_{i}=\frac{1-\sqrt{1-4m_{c}^{2}/s}}{2}, zi=ymc2ysmc2subscript𝑧𝑖𝑦superscriptsubscript𝑚𝑐2𝑦𝑠superscriptsubscript𝑚𝑐2z_{i}=\frac{ym_{c}^{2}}{ys-m_{c}^{2}}, m¯c2=(y+z)mc2yzsuperscriptsubscript¯𝑚𝑐2𝑦𝑧superscriptsubscript𝑚𝑐2𝑦𝑧\overline{m}_{c}^{2}=\frac{(y+z)m_{c}^{2}}{yz}, m~c2=mc2y(1y)superscriptsubscript~𝑚𝑐2superscriptsubscript𝑚𝑐2𝑦1𝑦\widetilde{m}_{c}^{2}=\frac{m_{c}^{2}}{y(1-y)}, yiyf𝑑y01𝑑ysuperscriptsubscriptsubscript𝑦𝑖subscript𝑦𝑓differential-d𝑦superscriptsubscript01differential-d𝑦\int_{y_{i}}^{y_{f}}dy\to\int_{0}^{1}dy, zi1y𝑑z01y𝑑zsuperscriptsubscriptsubscript𝑧𝑖1𝑦differential-d𝑧superscriptsubscript01𝑦differential-d𝑧\int_{z_{i}}^{1-y}dz\to\int_{0}^{1-y}dz when the δ𝛿\delta functions δ(sm¯c2)𝛿𝑠superscriptsubscript¯𝑚𝑐2\delta\left(s-\overline{m}_{c}^{2}\right) and δ(sm~c2)𝛿𝑠superscriptsubscript~𝑚𝑐2\delta\left(s-\widetilde{m}_{c}^{2}\right) appear.

Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number 11375063.

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